Higher differentiability of usual operations #
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We prove that the usual operations (addition, multiplication, difference, composition, and
so on) preserve C^n functions. We also expand the API around C^n functions.
Main results #
cont_diff.compstates that the composition of twoC^nfunctions isC^n.norm_iterated_fderiv_comp_legives the boundn! * C * D ^ nfor then-th derivative ofg ∘ fassuming that the derivatives ofgare bounded byCand thei-th derivative offis bounded byD ^ i.
Similar results are given for C^n functions on domains.
Notations #
We use the notation E [×n]→L[𝕜] F for the space of continuous multilinear maps on E^n with
values in F. This is the space in which the n-th derivative of a function from E to F lives.
In this file, we denote ⊤ : ℕ∞ with ∞.
Tags #
derivative, differentiability, higher derivative, C^n, multilinear, Taylor series, formal series
Constants #
@[simp]
theorem
iterated_fderiv_zero_fun
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ} :
iterated_fderiv 𝕜 n (λ (x : E), 0) = 0
theorem
cont_diff_zero_fun
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞} :
cont_diff 𝕜 n (λ (x : E), 0)
theorem
cont_diff_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{c : F} :
cont_diff 𝕜 n (λ (x : E), c)
Constants are C^∞.
theorem
cont_diff_on_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{c : F}
{s : set E} :
cont_diff_on 𝕜 n (λ (x : E), c) s
theorem
cont_diff_at_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{x : E}
{n : ℕ∞}
{c : F} :
cont_diff_at 𝕜 n (λ (x : E), c) x
theorem
cont_diff_within_at_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{s : set E}
{x : E}
{n : ℕ∞}
{c : F} :
cont_diff_within_at 𝕜 n (λ (x : E), c) s x
theorem
cont_diff_of_subsingleton
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{f : E → F}
{n : ℕ∞}
[subsingleton F] :
cont_diff 𝕜 n f
theorem
cont_diff_at_of_subsingleton
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{f : E → F}
{x : E}
{n : ℕ∞}
[subsingleton F] :
cont_diff_at 𝕜 n f x
theorem
cont_diff_within_at_of_subsingleton
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{s : set E}
{f : E → F}
{x : E}
{n : ℕ∞}
[subsingleton F] :
cont_diff_within_at 𝕜 n f s x
theorem
cont_diff_on_of_subsingleton
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{s : set E}
{f : E → F}
{n : ℕ∞}
[subsingleton F] :
cont_diff_on 𝕜 n f s
theorem
iterated_fderiv_succ_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
(n : ℕ)
(c : F) :
iterated_fderiv 𝕜 (n + 1) (λ (y : E), c) = 0
theorem
iterated_fderiv_const_of_ne
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ}
(hn : n ≠ 0)
(c : F) :
iterated_fderiv 𝕜 n (λ (y : E), c) = 0
Smoothness of linear functions #
theorem
is_bounded_linear_map.cont_diff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{f : E → F}
{n : ℕ∞}
(hf : is_bounded_linear_map 𝕜 f) :
cont_diff 𝕜 n f
Unbundled bounded linear functions are C^∞.
theorem
continuous_linear_map.cont_diff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
(f : E →L[𝕜] F) :
theorem
continuous_linear_equiv.cont_diff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
(f : E ≃L[𝕜] F) :
theorem
linear_isometry.cont_diff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
(f : E →ₗᵢ[𝕜] F) :
theorem
linear_isometry_equiv.cont_diff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
(f : E ≃ₗᵢ[𝕜] F) :
theorem
cont_diff_id
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{n : ℕ∞} :
The identity is C^∞.
theorem
cont_diff_within_at_id
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{n : ℕ∞}
{s : set E}
{x : E} :
cont_diff_within_at 𝕜 n id s x
theorem
cont_diff_at_id
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{n : ℕ∞}
{x : E} :
cont_diff_at 𝕜 n id x
theorem
cont_diff_on_id
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{n : ℕ∞}
{s : set E} :
cont_diff_on 𝕜 n id s
theorem
is_bounded_bilinear_map.cont_diff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{b : E × F → G}
{n : ℕ∞}
(hb : is_bounded_bilinear_map 𝕜 b) :
cont_diff 𝕜 n b
Bilinear functions are C^∞.
theorem
has_ftaylor_series_up_to_on.continuous_linear_map_comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{f : E → F}
{n : ℕ∞}
{p : E → formal_multilinear_series 𝕜 E F}
(g : F →L[𝕜] G)
(hf : has_ftaylor_series_up_to_on n f p s) :
has_ftaylor_series_up_to_on n (⇑g ∘ f) (λ (x : E) (k : ℕ), g.comp_continuous_multilinear_map (p x k)) s
If f admits a Taylor series p in a set s, and g is linear, then g ∘ f admits a Taylor
series whose k-th term is given by g ∘ (p k).
theorem
cont_diff_within_at.continuous_linear_map_comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{f : E → F}
{x : E}
{n : ℕ∞}
(g : F →L[𝕜] G)
(hf : cont_diff_within_at 𝕜 n f s x) :
cont_diff_within_at 𝕜 n (⇑g ∘ f) s x
Composition by continuous linear maps on the left preserves C^n functions in a domain
at a point.
theorem
cont_diff_at.continuous_linear_map_comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{f : E → F}
{x : E}
{n : ℕ∞}
(g : F →L[𝕜] G)
(hf : cont_diff_at 𝕜 n f x) :
cont_diff_at 𝕜 n (⇑g ∘ f) x
Composition by continuous linear maps on the left preserves C^n functions in a domain
at a point.
theorem
cont_diff_on.continuous_linear_map_comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{f : E → F}
{n : ℕ∞}
(g : F →L[𝕜] G)
(hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (⇑g ∘ f) s
Composition by continuous linear maps on the left preserves C^n functions on domains.
theorem
cont_diff.continuous_linear_map_comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : E → F}
(g : F →L[𝕜] G)
(hf : cont_diff 𝕜 n f) :
Composition by continuous linear maps on the left preserves C^n functions.
theorem
continuous_linear_map.iterated_fderiv_within_comp_left
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{x : E}
{n : ℕ∞}
{f : E → F}
(g : F →L[𝕜] G)
(hf : cont_diff_on 𝕜 n f s)
(hs : unique_diff_on 𝕜 s)
(hx : x ∈ s)
{i : ℕ}
(hi : ↑i ≤ n) :
iterated_fderiv_within 𝕜 i (⇑g ∘ f) s x = g.comp_continuous_multilinear_map (iterated_fderiv_within 𝕜 i f s x)
The iterated derivative within a set of the composition with a linear map on the left is obtained by applying the linear map to the iterated derivative.
theorem
continuous_linear_map.iterated_fderiv_comp_left
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : E → F}
(g : F →L[𝕜] G)
(hf : cont_diff 𝕜 n f)
(x : E)
{i : ℕ}
(hi : ↑i ≤ n) :
iterated_fderiv 𝕜 i (⇑g ∘ f) x = g.comp_continuous_multilinear_map (iterated_fderiv 𝕜 i f x)
The iterated derivative of the composition with a linear map on the left is obtained by applying the linear map to the iterated derivative.
theorem
continuous_linear_equiv.iterated_fderiv_within_comp_left
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{x : E}
(g : F ≃L[𝕜] G)
(f : E → F)
(hs : unique_diff_on 𝕜 s)
(hx : x ∈ s)
(i : ℕ) :
iterated_fderiv_within 𝕜 i (⇑g ∘ f) s x = ↑g.comp_continuous_multilinear_map (iterated_fderiv_within 𝕜 i f s x)
The iterated derivative within a set of the composition with a linear equiv on the left is obtained by applying the linear equiv to the iterated derivative. This is true without differentiability assumptions.
theorem
linear_isometry.norm_iterated_fderiv_within_comp_left
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{x : E}
{n : ℕ∞}
{f : E → F}
(g : F →ₗᵢ[𝕜] G)
(hf : cont_diff_on 𝕜 n f s)
(hs : unique_diff_on 𝕜 s)
(hx : x ∈ s)
{i : ℕ}
(hi : ↑i ≤ n) :
‖iterated_fderiv_within 𝕜 i (⇑g ∘ f) s x‖ = ‖iterated_fderiv_within 𝕜 i f s x‖
Composition with a linear isometry on the left preserves the norm of the iterated derivative within a set.
theorem
linear_isometry.norm_iterated_fderiv_comp_left
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : E → F}
(g : F →ₗᵢ[𝕜] G)
(hf : cont_diff 𝕜 n f)
(x : E)
{i : ℕ}
(hi : ↑i ≤ n) :
‖iterated_fderiv 𝕜 i (⇑g ∘ f) x‖ = ‖iterated_fderiv 𝕜 i f x‖
Composition with a linear isometry on the left preserves the norm of the iterated derivative.
theorem
linear_isometry_equiv.norm_iterated_fderiv_within_comp_left
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{x : E}
(g : F ≃ₗᵢ[𝕜] G)
(f : E → F)
(hs : unique_diff_on 𝕜 s)
(hx : x ∈ s)
(i : ℕ) :
‖iterated_fderiv_within 𝕜 i (⇑g ∘ f) s x‖ = ‖iterated_fderiv_within 𝕜 i f s x‖
Composition with a linear isometry equiv on the left preserves the norm of the iterated derivative within a set.
theorem
linear_isometry_equiv.norm_iterated_fderiv_comp_left
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
(g : F ≃ₗᵢ[𝕜] G)
(f : E → F)
(x : E)
(i : ℕ) :
‖iterated_fderiv 𝕜 i (⇑g ∘ f) x‖ = ‖iterated_fderiv 𝕜 i f x‖
Composition with a linear isometry equiv on the left preserves the norm of the iterated derivative.
theorem
continuous_linear_equiv.comp_cont_diff_within_at_iff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{f : E → F}
{x : E}
{n : ℕ∞}
(e : F ≃L[𝕜] G) :
cont_diff_within_at 𝕜 n (⇑e ∘ f) s x ↔ cont_diff_within_at 𝕜 n f s x
Composition by continuous linear equivs on the left respects higher differentiability at a point in a domain.
theorem
continuous_linear_equiv.comp_cont_diff_at_iff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{f : E → F}
{x : E}
{n : ℕ∞}
(e : F ≃L[𝕜] G) :
cont_diff_at 𝕜 n (⇑e ∘ f) x ↔ cont_diff_at 𝕜 n f x
Composition by continuous linear equivs on the left respects higher differentiability at a point.
theorem
continuous_linear_equiv.comp_cont_diff_on_iff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{f : E → F}
{n : ℕ∞}
(e : F ≃L[𝕜] G) :
cont_diff_on 𝕜 n (⇑e ∘ f) s ↔ cont_diff_on 𝕜 n f s
Composition by continuous linear equivs on the left respects higher differentiability on domains.
theorem
continuous_linear_equiv.comp_cont_diff_iff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{f : E → F}
{n : ℕ∞}
(e : F ≃L[𝕜] G) :
Composition by continuous linear equivs on the left respects higher differentiability.
theorem
has_ftaylor_series_up_to_on.comp_continuous_linear_map
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{f : E → F}
{n : ℕ∞}
{p : E → formal_multilinear_series 𝕜 E F}
(hf : has_ftaylor_series_up_to_on n f p s)
(g : G →L[𝕜] E) :
has_ftaylor_series_up_to_on n (f ∘ ⇑g) (λ (x : G) (k : ℕ), (p (⇑g x) k).comp_continuous_linear_map (λ (_x : fin k), g)) (⇑g ⁻¹' s)
If f admits a Taylor series p in a set s, and g is linear, then f ∘ g admits a Taylor
series in g ⁻¹' s, whose k-th term is given by p k (g v₁, ..., g vₖ) .
theorem
cont_diff_within_at.comp_continuous_linear_map
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{f : E → F}
{n : ℕ∞}
{x : G}
(g : G →L[𝕜] E)
(hf : cont_diff_within_at 𝕜 n f s (⇑g x)) :
cont_diff_within_at 𝕜 n (f ∘ ⇑g) (⇑g ⁻¹' s) x
Composition by continuous linear maps on the right preserves C^n functions at a point on
a domain.
theorem
cont_diff_on.comp_continuous_linear_map
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{f : E → F}
{n : ℕ∞}
(hf : cont_diff_on 𝕜 n f s)
(g : G →L[𝕜] E) :
cont_diff_on 𝕜 n (f ∘ ⇑g) (⇑g ⁻¹' s)
Composition by continuous linear maps on the right preserves C^n functions on domains.
theorem
cont_diff.comp_continuous_linear_map
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : E → F}
{g : G →L[𝕜] E}
(hf : cont_diff 𝕜 n f) :
Composition by continuous linear maps on the right preserves C^n functions.
theorem
continuous_linear_map.iterated_fderiv_within_comp_right
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{n : ℕ∞}
{f : E → F}
(g : G →L[𝕜] E)
(hf : cont_diff_on 𝕜 n f s)
(hs : unique_diff_on 𝕜 s)
(h's : unique_diff_on 𝕜 (⇑g ⁻¹' s))
{x : G}
(hx : ⇑g x ∈ s)
{i : ℕ}
(hi : ↑i ≤ n) :
iterated_fderiv_within 𝕜 i (f ∘ ⇑g) (⇑g ⁻¹' s) x = (iterated_fderiv_within 𝕜 i f s (⇑g x)).comp_continuous_linear_map (λ (_x : fin i), g)
The iterated derivative within a set of the composition with a linear map on the right is obtained by composing the iterated derivative with the linear map.
theorem
continuous_linear_equiv.iterated_fderiv_within_comp_right
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
(g : G ≃L[𝕜] E)
(f : E → F)
(hs : unique_diff_on 𝕜 s)
{x : G}
(hx : ⇑g x ∈ s)
(i : ℕ) :
iterated_fderiv_within 𝕜 i (f ∘ ⇑g) (⇑g ⁻¹' s) x = (iterated_fderiv_within 𝕜 i f s (⇑g x)).comp_continuous_linear_map (λ (_x : fin i), ↑g)
The iterated derivative within a set of the composition with a linear equiv on the right is obtained by composing the iterated derivative with the linear equiv.
theorem
continuous_linear_map.iterated_fderiv_comp_right
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
(g : G →L[𝕜] E)
{f : E → F}
(hf : cont_diff 𝕜 n f)
(x : G)
{i : ℕ}
(hi : ↑i ≤ n) :
iterated_fderiv 𝕜 i (f ∘ ⇑g) x = (iterated_fderiv 𝕜 i f (⇑g x)).comp_continuous_linear_map (λ (_x : fin i), g)
The iterated derivative of the composition with a linear map on the right is obtained by composing the iterated derivative with the linear map.
theorem
linear_isometry_equiv.norm_iterated_fderiv_within_comp_right
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
(g : G ≃ₗᵢ[𝕜] E)
(f : E → F)
(hs : unique_diff_on 𝕜 s)
{x : G}
(hx : ⇑g x ∈ s)
(i : ℕ) :
Composition with a linear isometry on the right preserves the norm of the iterated derivative within a set.
theorem
linear_isometry_equiv.norm_iterated_fderiv_comp_right
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
(g : G ≃ₗᵢ[𝕜] E)
(f : E → F)
(x : G)
(i : ℕ) :
‖iterated_fderiv 𝕜 i (f ∘ ⇑g) x‖ = ‖iterated_fderiv 𝕜 i f (⇑g x)‖
Composition with a linear isometry on the right preserves the norm of the iterated derivative within a set.
theorem
continuous_linear_equiv.cont_diff_within_at_comp_iff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{f : E → F}
{x : E}
{n : ℕ∞}
(e : G ≃L[𝕜] E) :
cont_diff_within_at 𝕜 n (f ∘ ⇑e) (⇑e ⁻¹' s) (⇑(e.symm) x) ↔ cont_diff_within_at 𝕜 n f s x
Composition by continuous linear equivs on the right respects higher differentiability at a point in a domain.
theorem
continuous_linear_equiv.cont_diff_at_comp_iff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{f : E → F}
{x : E}
{n : ℕ∞}
(e : G ≃L[𝕜] E) :
cont_diff_at 𝕜 n (f ∘ ⇑e) (⇑(e.symm) x) ↔ cont_diff_at 𝕜 n f x
Composition by continuous linear equivs on the right respects higher differentiability at a point.
theorem
continuous_linear_equiv.cont_diff_on_comp_iff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{f : E → F}
{n : ℕ∞}
(e : G ≃L[𝕜] E) :
cont_diff_on 𝕜 n (f ∘ ⇑e) (⇑e ⁻¹' s) ↔ cont_diff_on 𝕜 n f s
Composition by continuous linear equivs on the right respects higher differentiability on domains.
theorem
continuous_linear_equiv.cont_diff_comp_iff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{f : E → F}
{n : ℕ∞}
(e : G ≃L[𝕜] E) :
Composition by continuous linear equivs on the right respects higher differentiability.
theorem
has_ftaylor_series_up_to_on.prod
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{f : E → F}
{n : ℕ∞}
{p : E → formal_multilinear_series 𝕜 E F}
(hf : has_ftaylor_series_up_to_on n f p s)
{g : E → G}
{q : E → formal_multilinear_series 𝕜 E G}
(hg : has_ftaylor_series_up_to_on n g q s) :
has_ftaylor_series_up_to_on n (λ (y : E), (f y, g y)) (λ (y : E) (k : ℕ), (p y k).prod (q y k)) s
If two functions f and g admit Taylor series p and q in a set s, then the cartesian
product of f and g admits the cartesian product of p and q as a Taylor series.
theorem
cont_diff_within_at.prod
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{x : E}
{n : ℕ∞}
{s : set E}
{f : E → F}
{g : E → G}
(hf : cont_diff_within_at 𝕜 n f s x)
(hg : cont_diff_within_at 𝕜 n g s x) :
cont_diff_within_at 𝕜 n (λ (x : E), (f x, g x)) s x
The cartesian product of C^n functions at a point in a domain is C^n.
theorem
cont_diff_on.prod
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{s : set E}
{f : E → F}
{g : E → G}
(hf : cont_diff_on 𝕜 n f s)
(hg : cont_diff_on 𝕜 n g s) :
cont_diff_on 𝕜 n (λ (x : E), (f x, g x)) s
The cartesian product of C^n functions on domains is C^n.
theorem
cont_diff_at.prod
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{x : E}
{n : ℕ∞}
{f : E → F}
{g : E → G}
(hf : cont_diff_at 𝕜 n f x)
(hg : cont_diff_at 𝕜 n g x) :
cont_diff_at 𝕜 n (λ (x : E), (f x, g x)) x
The cartesian product of C^n functions at a point is C^n.
theorem
cont_diff.prod
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : E → F}
{g : E → G}
(hf : cont_diff 𝕜 n f)
(hg : cont_diff 𝕜 n g) :
cont_diff 𝕜 n (λ (x : E), (f x, g x))
The cartesian product of C^n functions is C^n.
Composition of C^n functions #
We show that the composition of C^n functions is C^n. One way to prove it would be to write
the n-th derivative of the composition (this is Faà di Bruno's formula) and check its continuity,
but this is very painful. Instead, we go for a simple inductive proof. Assume it is done for n.
Then, to check it for n+1, one needs to check that the derivative of g ∘ f is C^n, i.e.,
that Dg(f x) ⬝ Df(x) is C^n. The term Dg (f x) is the composition of two C^n functions, so
it is C^n by the inductive assumption. The term Df(x) is also C^n. Then, the matrix
multiplication is the application of a bilinear map (which is C^∞, and therefore C^n) to
x ↦ (Dg(f x), Df x). As the composition of two C^n maps, it is again C^n, and we are done.
There is a subtlety in this argument: we apply the inductive assumption to functions on other Banach
spaces. In maths, one would say: prove by induction over n that, for all C^n maps between all
pairs of Banach spaces, their composition is C^n. In Lean, this is fine as long as the spaces
stay in the same universe. This is not the case in the above argument: if E lives in universe u
and F lives in universe v, then linear maps from E to F (to which the derivative of f
belongs) is in universe max u v. If one could quantify over finitely many universes, the above
proof would work fine, but this is not the case. One could still write the proof considering spaces
in any universe in u, v, w, max u v, max v w, max u v w, but it would be extremely tedious and
lead to a lot of duplication. Instead, we formulate the above proof when all spaces live in the same
universe (where everything is fine), and then we deduce the general result by lifting all our spaces
to a common universe. We use the trick that any space H is isomorphic through a continuous linear
equiv to continuous_multilinear_map (λ (i : fin 0), E × F × G) H to change the universe level,
and then argue that composing with such a linear equiv does not change the fact of being C^n,
which we have already proved previously.
theorem
cont_diff_on.comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{s : set E}
{t : set F}
{g : F → G}
{f : E → F}
(hg : cont_diff_on 𝕜 n g t)
(hf : cont_diff_on 𝕜 n f s)
(st : s ⊆ f ⁻¹' t) :
cont_diff_on 𝕜 n (g ∘ f) s
The composition of C^n functions on domains is C^n.
theorem
cont_diff_on.comp'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{s : set E}
{t : set F}
{g : F → G}
{f : E → F}
(hg : cont_diff_on 𝕜 n g t)
(hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (g ∘ f) (s ∩ f ⁻¹' t)
The composition of C^n functions on domains is C^n.
theorem
cont_diff.comp_cont_diff_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{s : set E}
{g : F → G}
{f : E → F}
(hg : cont_diff 𝕜 n g)
(hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (g ∘ f) s
The composition of a C^n function on a domain with a C^n function is C^n.
theorem
cont_diff.comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{g : F → G}
{f : E → F}
(hg : cont_diff 𝕜 n g)
(hf : cont_diff 𝕜 n f) :
The composition of C^n functions is C^n.
theorem
cont_diff_within_at.comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{s : set E}
{t : set F}
{g : F → G}
{f : E → F}
(x : E)
(hg : cont_diff_within_at 𝕜 n g t (f x))
(hf : cont_diff_within_at 𝕜 n f s x)
(st : s ⊆ f ⁻¹' t) :
cont_diff_within_at 𝕜 n (g ∘ f) s x
The composition of C^n functions at points in domains is C^n.
theorem
cont_diff_within_at.comp_of_mem
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{s : set E}
{t : set F}
{g : F → G}
{f : E → F}
(x : E)
(hg : cont_diff_within_at 𝕜 n g t (f x))
(hf : cont_diff_within_at 𝕜 n f s x)
(hs : t ∈ nhds_within (f x) (f '' s)) :
cont_diff_within_at 𝕜 n (g ∘ f) s x
The composition of C^n functions at points in domains is C^n,
with a weaker condition on s and t.
theorem
cont_diff_within_at.comp'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{s : set E}
{t : set F}
{g : F → G}
{f : E → F}
(x : E)
(hg : cont_diff_within_at 𝕜 n g t (f x))
(hf : cont_diff_within_at 𝕜 n f s x) :
cont_diff_within_at 𝕜 n (g ∘ f) (s ∩ f ⁻¹' t) x
The composition of C^n functions at points in domains is C^n.
theorem
cont_diff_at.comp_cont_diff_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{f : E → F}
{g : F → G}
{n : ℕ∞}
(x : E)
(hg : cont_diff_at 𝕜 n g (f x))
(hf : cont_diff_within_at 𝕜 n f s x) :
cont_diff_within_at 𝕜 n (g ∘ f) s x
theorem
cont_diff_at.comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{f : E → F}
{g : F → G}
{n : ℕ∞}
(x : E)
(hg : cont_diff_at 𝕜 n g (f x))
(hf : cont_diff_at 𝕜 n f x) :
cont_diff_at 𝕜 n (g ∘ f) x
The composition of C^n functions at points is C^n.
theorem
cont_diff.comp_cont_diff_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{t : set E}
{x : E}
{n : ℕ∞}
{g : F → G}
{f : E → F}
(h : cont_diff 𝕜 n g)
(hf : cont_diff_within_at 𝕜 n f t x) :
cont_diff_within_at 𝕜 n (g ∘ f) t x
theorem
cont_diff.comp_cont_diff_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{g : F → G}
{f : E → F}
(x : E)
(hg : cont_diff 𝕜 n g)
(hf : cont_diff_at 𝕜 n f x) :
cont_diff_at 𝕜 n (g ∘ f) x
Smoothness of projections #
theorem
cont_diff_fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞} :
The first projection in a product is C^∞.
theorem
cont_diff.fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : E → F × G}
(hf : cont_diff 𝕜 n f) :
Postcomposing f with prod.fst is C^n
theorem
cont_diff.fst'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : E → G}
(hf : cont_diff 𝕜 n f) :
Precomposing f with prod.fst is C^n
theorem
cont_diff_on_fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{s : set (E × F)} :
cont_diff_on 𝕜 n prod.fst s
The first projection on a domain in a product is C^∞.
theorem
cont_diff_on.fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : E → F × G}
{s : set E}
(hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (λ (x : E), (f x).fst) s
theorem
cont_diff_at_fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{p : E × F} :
cont_diff_at 𝕜 n prod.fst p
The first projection at a point in a product is C^∞.
theorem
cont_diff_at.fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : E → F × G}
{x : E}
(hf : cont_diff_at 𝕜 n f x) :
cont_diff_at 𝕜 n (λ (x : E), (f x).fst) x
Postcomposing f with prod.fst is C^n at (x, y)
theorem
cont_diff_at.fst'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : E → G}
{x : E}
{y : F}
(hf : cont_diff_at 𝕜 n f x) :
cont_diff_at 𝕜 n (λ (x : E × F), f x.fst) (x, y)
Precomposing f with prod.fst is C^n at (x, y)
theorem
cont_diff_at.fst''
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : E → G}
{x : E × F}
(hf : cont_diff_at 𝕜 n f x.fst) :
cont_diff_at 𝕜 n (λ (x : E × F), f x.fst) x
Precomposing f with prod.fst is C^n at x : E × F
theorem
cont_diff_within_at_fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{s : set (E × F)}
{p : E × F} :
cont_diff_within_at 𝕜 n prod.fst s p
The first projection within a domain at a point in a product is C^∞.
theorem
cont_diff_snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞} :
The second projection in a product is C^∞.
theorem
cont_diff.snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : E → F × G}
(hf : cont_diff 𝕜 n f) :
Postcomposing f with prod.snd is C^n
theorem
cont_diff.snd'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : F → G}
(hf : cont_diff 𝕜 n f) :
Precomposing f with prod.snd is C^n
theorem
cont_diff_on_snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{s : set (E × F)} :
cont_diff_on 𝕜 n prod.snd s
The second projection on a domain in a product is C^∞.
theorem
cont_diff_on.snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : E → F × G}
{s : set E}
(hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (λ (x : E), (f x).snd) s
theorem
cont_diff_at_snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{p : E × F} :
cont_diff_at 𝕜 n prod.snd p
The second projection at a point in a product is C^∞.
theorem
cont_diff_at.snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : E → F × G}
{x : E}
(hf : cont_diff_at 𝕜 n f x) :
cont_diff_at 𝕜 n (λ (x : E), (f x).snd) x
Postcomposing f with prod.snd is C^n at x
theorem
cont_diff_at.snd'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : F → G}
{x : E}
{y : F}
(hf : cont_diff_at 𝕜 n f y) :
cont_diff_at 𝕜 n (λ (x : E × F), f x.snd) (x, y)
Precomposing f with prod.snd is C^n at (x, y)
theorem
cont_diff_at.snd''
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{f : F → G}
{x : E × F}
(hf : cont_diff_at 𝕜 n f x.snd) :
cont_diff_at 𝕜 n (λ (x : E × F), f x.snd) x
Precomposing f with prod.snd is C^n at x : E × F
theorem
cont_diff_within_at_snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{s : set (E × F)}
{p : E × F} :
cont_diff_within_at 𝕜 n prod.snd s p
The second projection within a domain at a point in a product is C^∞.
theorem
cont_diff.comp₂
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{E₁ : Type u_3}
{E₂ : Type u_4}
[normed_add_comm_group E₁]
[normed_add_comm_group E₂]
[normed_space 𝕜 E₁]
[normed_space 𝕜 E₂]
{g : E₁ × E₂ → G}
{f₁ : F → E₁}
{f₂ : F → E₂}
(hg : cont_diff 𝕜 n g)
(hf₁ : cont_diff 𝕜 n f₁)
(hf₂ : cont_diff 𝕜 n f₂) :
cont_diff 𝕜 n (λ (x : F), g (f₁ x, f₂ x))
theorem
cont_diff.comp₃
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{E₁ : Type u_3}
{E₂ : Type u_4}
{E₃ : Type u_5}
[normed_add_comm_group E₁]
[normed_add_comm_group E₂]
[normed_add_comm_group E₃]
[normed_space 𝕜 E₁]
[normed_space 𝕜 E₂]
[normed_space 𝕜 E₃]
{g : E₁ × E₂ × E₃ → G}
{f₁ : F → E₁}
{f₂ : F → E₂}
{f₃ : F → E₃}
(hg : cont_diff 𝕜 n g)
(hf₁ : cont_diff 𝕜 n f₁)
(hf₂ : cont_diff 𝕜 n f₂)
(hf₃ : cont_diff 𝕜 n f₃) :
cont_diff 𝕜 n (λ (x : F), g (f₁ x, f₂ x, f₃ x))
theorem
cont_diff.comp_cont_diff_on₂
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{E₁ : Type u_3}
{E₂ : Type u_4}
[normed_add_comm_group E₁]
[normed_add_comm_group E₂]
[normed_space 𝕜 E₁]
[normed_space 𝕜 E₂]
{g : E₁ × E₂ → G}
{f₁ : F → E₁}
{f₂ : F → E₂}
{s : set F}
(hg : cont_diff 𝕜 n g)
(hf₁ : cont_diff_on 𝕜 n f₁ s)
(hf₂ : cont_diff_on 𝕜 n f₂ s) :
cont_diff_on 𝕜 n (λ (x : F), g (f₁ x, f₂ x)) s
theorem
cont_diff.comp_cont_diff_on₃
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{n : ℕ∞}
{E₁ : Type u_3}
{E₂ : Type u_4}
{E₃ : Type u_5}
[normed_add_comm_group E₁]
[normed_add_comm_group E₂]
[normed_add_comm_group E₃]
[normed_space 𝕜 E₁]
[normed_space 𝕜 E₂]
[normed_space 𝕜 E₃]
{g : E₁ × E₂ × E₃ → G}
{f₁ : F → E₁}
{f₂ : F → E₂}
{f₃ : F → E₃}
{s : set F}
(hg : cont_diff 𝕜 n g)
(hf₁ : cont_diff_on 𝕜 n f₁ s)
(hf₂ : cont_diff_on 𝕜 n f₂ s)
(hf₃ : cont_diff_on 𝕜 n f₃ s) :
cont_diff_on 𝕜 n (λ (x : F), g (f₁ x, f₂ x, f₃ x)) s
theorem
cont_diff.clm_comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{X : Type u_2}
[normed_add_comm_group X]
[normed_space 𝕜 X]
{n : ℕ∞}
{g : X → (F →L[𝕜] G)}
{f : X → (E →L[𝕜] F)}
(hg : cont_diff 𝕜 n g)
(hf : cont_diff 𝕜 n f) :
theorem
cont_diff_on.clm_comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{X : Type u_2}
[normed_add_comm_group X]
[normed_space 𝕜 X]
{n : ℕ∞}
{g : X → (F →L[𝕜] G)}
{f : X → (E →L[𝕜] F)}
{s : set X}
(hg : cont_diff_on 𝕜 n g s)
(hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (λ (x : X), (g x).comp (f x)) s
theorem
cont_diff.clm_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{f : E → (F →L[𝕜] G)}
{g : E → F}
{n : ℕ∞}
(hf : cont_diff 𝕜 n f)
(hg : cont_diff 𝕜 n g) :
theorem
cont_diff_on.clm_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{f : E → (F →L[𝕜] G)}
{g : E → F}
{n : ℕ∞}
(hf : cont_diff_on 𝕜 n f s)
(hg : cont_diff_on 𝕜 n g s) :
cont_diff_on 𝕜 n (λ (x : E), ⇑(f x) (g x)) s
theorem
cont_diff.smul_right
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{f : E → (F →L[𝕜] 𝕜)}
{g : E → G}
{n : ℕ∞}
(hf : cont_diff 𝕜 n f)
(hg : cont_diff 𝕜 n g) :
cont_diff 𝕜 n (λ (x : E), (f x).smul_right (g x))
theorem
cont_diff_prod_assoc
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G] :
cont_diff 𝕜 ⊤ ⇑(equiv.prod_assoc E F G)
The natural equivalence (E × F) × G ≃ E × (F × G) is smooth.
Warning: if you think you need this lemma, it is likely that you can simplify your proof by reformulating the lemma that you're applying next using the tips in Note [continuity lemma statement]
theorem
cont_diff_prod_assoc_symm
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G] :
cont_diff 𝕜 ⊤ ⇑((equiv.prod_assoc E F G).symm)
The natural equivalence E × (F × G) ≃ (E × F) × G is smooth.
Warning: see remarks attached to cont_diff_prod_assoc
Bundled derivatives are smooth #
theorem
cont_diff_within_at.has_fderiv_within_at_nhds
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{f : E → F → G}
{g : E → F}
{t : set F}
{n : ℕ}
{x₀ : E}
(hf : cont_diff_within_at 𝕜 (↑n + 1) (function.uncurry f) (has_insert.insert x₀ s ×ˢ t) (x₀, g x₀))
(hg : cont_diff_within_at 𝕜 ↑n g s x₀)
(hgt : t ∈ nhds_within (g x₀) (g '' s)) :
∃ (v : set E) (H : v ∈ nhds_within x₀ (has_insert.insert x₀ s)), v ⊆ has_insert.insert x₀ s ∧ ∃ (f' : E → (F →L[𝕜] G)), (∀ (x : E), x ∈ v → has_fderiv_within_at (f x) (f' x) t (g x)) ∧ cont_diff_within_at 𝕜 ↑n (λ (x : E), f' x) s x₀
One direction of cont_diff_within_at_succ_iff_has_fderiv_within_at, but where all derivatives
are taken within the same set. Version for partial derivatives / functions with parameters.
If f x is a C^n+1 family of functions and g x is a C^n family of points, then the
derivative of f x at g x depends in a C^n way on x. We give a general version of this fact
relative to sets which may not have unique derivatives, in the following form.
If f : E × F → G is C^n+1 at (x₀, g(x₀)) in (s ∪ {x₀}) × t ⊆ E × F and g : E → F is
C^n at x₀ within some set s ⊆ E, then there is a function f' : E → F →L[𝕜] G
that is C^n at x₀ within s such that for all x sufficiently close to x₀ within
s ∪ {x₀} the function y ↦ f x y has derivative f' x at g x within t ⊆ F.
For convenience, we return an explicit set of x's where this holds that is a subset of
s ∪ {x₀}.
We need one additional condition, namely that t is a neighborhood of g(x₀) within g '' s.
theorem
cont_diff_within_at.fderiv_within''
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{x₀ : E}
{m : ℕ∞}
{f : E → F → G}
{g : E → F}
{t : set F}
{n : ℕ∞}
(hf : cont_diff_within_at 𝕜 n (function.uncurry f) (has_insert.insert x₀ s ×ˢ t) (x₀, g x₀))
(hg : cont_diff_within_at 𝕜 m g s x₀)
(ht : ∀ᶠ (x : E) in nhds_within x₀ (has_insert.insert x₀ s), unique_diff_within_at 𝕜 t (g x))
(hmn : m + 1 ≤ n)
(hgt : t ∈ nhds_within (g x₀) (g '' s)) :
cont_diff_within_at 𝕜 m (λ (x : E), fderiv_within 𝕜 (f x) t (g x)) s x₀
The most general lemma stating that x ↦ fderiv_within 𝕜 (f x) t (g x) is C^n
at a point within a set.
To show that x ↦ D_yf(x,y)g(x) (taken within t) is C^m at x₀ within s, we require that
fisC^nat(x₀, g(x₀))within(s ∪ {x₀}) × tforn ≥ m+1.gisC^matx₀withins;- Derivatives are unique at
g(x)withintforxsufficiently close tox₀withins ∪ {x₀}; tis a neighborhood ofg(x₀)withing '' s;
theorem
cont_diff_within_at.fderiv_within'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{x₀ : E}
{m : ℕ∞}
{f : E → F → G}
{g : E → F}
{t : set F}
{n : ℕ∞}
(hf : cont_diff_within_at 𝕜 n (function.uncurry f) (has_insert.insert x₀ s ×ˢ t) (x₀, g x₀))
(hg : cont_diff_within_at 𝕜 m g s x₀)
(ht : ∀ᶠ (x : E) in nhds_within x₀ (has_insert.insert x₀ s), unique_diff_within_at 𝕜 t (g x))
(hmn : m + 1 ≤ n)
(hst : s ⊆ g ⁻¹' t) :
cont_diff_within_at 𝕜 m (λ (x : E), fderiv_within 𝕜 (f x) t (g x)) s x₀
A special case of cont_diff_within_at.fderiv_within'' where we require that s ⊆ g⁻¹(t).
theorem
cont_diff_within_at.fderiv_within
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{x₀ : E}
{m : ℕ∞}
{f : E → F → G}
{g : E → F}
{t : set F}
{n : ℕ∞}
(hf : cont_diff_within_at 𝕜 n (function.uncurry f) (s ×ˢ t) (x₀, g x₀))
(hg : cont_diff_within_at 𝕜 m g s x₀)
(ht : unique_diff_on 𝕜 t)
(hmn : m + 1 ≤ n)
(hx₀ : x₀ ∈ s)
(hst : s ⊆ g ⁻¹' t) :
cont_diff_within_at 𝕜 m (λ (x : E), fderiv_within 𝕜 (f x) t (g x)) s x₀
A special case of cont_diff_within_at.fderiv_within' where we require that x₀ ∈ s and there
are unique derivatives everywhere within t.
theorem
cont_diff_within_at.fderiv_within_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{s : set E}
{x₀ : E}
{m : ℕ∞}
{f : E → F → G}
{g k : E → F}
{t : set F}
{n : ℕ∞}
(hf : cont_diff_within_at 𝕜 n (function.uncurry f) (s ×ˢ t) (x₀, g x₀))
(hg : cont_diff_within_at 𝕜 m g s x₀)
(hk : cont_diff_within_at 𝕜 m k s x₀)
(ht : unique_diff_on 𝕜 t)
(hmn : m + 1 ≤ n)
(hx₀ : x₀ ∈ s)
(hst : s ⊆ g ⁻¹' t) :
cont_diff_within_at 𝕜 m (λ (x : E), ⇑(fderiv_within 𝕜 (f x) t (g x)) (k x)) s x₀
x ↦ fderiv_within 𝕜 (f x) t (g x) (k x) is smooth at a point within a set.
theorem
cont_diff_within_at.fderiv_within_right
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{s : set E}
{f : E → F}
{x₀ : E}
{m n : ℕ∞}
(hf : cont_diff_within_at 𝕜 n f s x₀)
(hs : unique_diff_on 𝕜 s)
(hmn : m + 1 ≤ n)
(hx₀s : x₀ ∈ s) :
cont_diff_within_at 𝕜 m (fderiv_within 𝕜 f s) s x₀
fderiv_within 𝕜 f s is smooth at x₀ within s.
theorem
cont_diff_at.fderiv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{x₀ : E}
{m : ℕ∞}
{f : E → F → G}
{g : E → F}
{n : ℕ∞}
(hf : cont_diff_at 𝕜 n (function.uncurry f) (x₀, g x₀))
(hg : cont_diff_at 𝕜 m g x₀)
(hmn : m + 1 ≤ n) :
cont_diff_at 𝕜 m (λ (x : E), fderiv 𝕜 (f x) (g x)) x₀
x ↦ fderiv 𝕜 (f x) (g x) is smooth at x₀.
theorem
cont_diff_at.fderiv_right
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{f : E → F}
{x₀ : E}
{m n : ℕ∞}
(hf : cont_diff_at 𝕜 n f x₀)
(hmn : m + 1 ≤ n) :
cont_diff_at 𝕜 m (fderiv 𝕜 f) x₀
fderiv 𝕜 f is smooth at x₀.
theorem
cont_diff.fderiv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{f : E → F → G}
{g : E → F}
{n m : ℕ∞}
(hf : cont_diff 𝕜 m (function.uncurry f))
(hg : cont_diff 𝕜 n g)
(hnm : n + 1 ≤ m) :
x ↦ fderiv 𝕜 (f x) (g x) is smooth.
theorem
cont_diff.fderiv_right
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{f : E → F}
{m n : ℕ∞}
(hf : cont_diff 𝕜 n f)
(hmn : m + 1 ≤ n) :
fderiv 𝕜 f is smooth.
theorem
continuous.fderiv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{f : E → F → G}
{g : E → F}
{n : ℕ∞}
(hf : cont_diff 𝕜 n (function.uncurry f))
(hg : continuous g)
(hn : 1 ≤ n) :
continuous (λ (x : E), fderiv 𝕜 (f x) (g x))
x ↦ fderiv 𝕜 (f x) (g x) is continuous.
theorem
cont_diff.fderiv_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{f : E → F → G}
{g k : E → F}
{n m : ℕ∞}
(hf : cont_diff 𝕜 m (function.uncurry f))
(hg : cont_diff 𝕜 n g)
(hk : cont_diff 𝕜 n k)
(hnm : n + 1 ≤ m) :
x ↦ fderiv 𝕜 (f x) (g x) (k x) is smooth.
theorem
cont_diff_on_fderiv_within_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{m n : ℕ∞}
{s : set E}
{f : E → F}
(hf : cont_diff_on 𝕜 n f s)
(hs : unique_diff_on 𝕜 s)
(hmn : m + 1 ≤ n) :
cont_diff_on 𝕜 m (λ (p : E × E), ⇑(fderiv_within 𝕜 f s p.fst) p.snd) (s ×ˢ set.univ)
The bundled derivative of a C^{n+1} function is C^n.
theorem
cont_diff_on.continuous_on_fderiv_within_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{s : set E}
{f : E → F}
{n : ℕ∞}
(hf : cont_diff_on 𝕜 n f s)
(hs : unique_diff_on 𝕜 s)
(hn : 1 ≤ n) :
continuous_on (λ (p : E × E), ⇑(fderiv_within 𝕜 f s p.fst) p.snd) (s ×ˢ set.univ)
If a function is at least C^1, its bundled derivative (mapping (x, v) to Df(x) v) is
continuous.
theorem
cont_diff.cont_diff_fderiv_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{m n : ℕ∞}
{f : E → F}
(hf : cont_diff 𝕜 n f)
(hmn : m + 1 ≤ n) :
The bundled derivative of a C^{n+1} function is C^n.
Smoothness of functions f : E → Π i, F' i #
theorem
has_ftaylor_series_up_to_on_pi
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{n : ℕ∞}
{ι : Type u_3}
[fintype ι]
{F' : ι → Type u_5}
[Π (i : ι), normed_add_comm_group (F' i)]
[Π (i : ι), normed_space 𝕜 (F' i)]
{φ : Π (i : ι), E → F' i}
{p' : Π (i : ι), E → formal_multilinear_series 𝕜 E (F' i)} :
has_ftaylor_series_up_to_on n (λ (x : E) (i : ι), φ i x) (λ (x : E) (m : ℕ), continuous_multilinear_map.pi (λ (i : ι), p' i x m)) s ↔ ∀ (i : ι), has_ftaylor_series_up_to_on n (φ i) (p' i) s
@[simp]
theorem
has_ftaylor_series_up_to_on_pi'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{n : ℕ∞}
{ι : Type u_3}
[fintype ι]
{F' : ι → Type u_5}
[Π (i : ι), normed_add_comm_group (F' i)]
[Π (i : ι), normed_space 𝕜 (F' i)]
{Φ : E → Π (i : ι), F' i}
{P' : E → formal_multilinear_series 𝕜 E (Π (i : ι), F' i)} :
has_ftaylor_series_up_to_on n Φ P' s ↔ ∀ (i : ι), has_ftaylor_series_up_to_on n (λ (x : E), Φ x i) (λ (x : E) (m : ℕ), (continuous_linear_map.proj i).comp_continuous_multilinear_map (P' x m)) s
theorem
cont_diff_within_at_pi
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{x : E}
{n : ℕ∞}
{ι : Type u_3}
[fintype ι]
{F' : ι → Type u_5}
[Π (i : ι), normed_add_comm_group (F' i)]
[Π (i : ι), normed_space 𝕜 (F' i)]
{Φ : E → Π (i : ι), F' i} :
cont_diff_within_at 𝕜 n Φ s x ↔ ∀ (i : ι), cont_diff_within_at 𝕜 n (λ (x : E), Φ x i) s x
theorem
cont_diff_on_pi
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{n : ℕ∞}
{ι : Type u_3}
[fintype ι]
{F' : ι → Type u_5}
[Π (i : ι), normed_add_comm_group (F' i)]
[Π (i : ι), normed_space 𝕜 (F' i)]
{Φ : E → Π (i : ι), F' i} :
cont_diff_on 𝕜 n Φ s ↔ ∀ (i : ι), cont_diff_on 𝕜 n (λ (x : E), Φ x i) s
theorem
cont_diff_at_pi
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{x : E}
{n : ℕ∞}
{ι : Type u_3}
[fintype ι]
{F' : ι → Type u_5}
[Π (i : ι), normed_add_comm_group (F' i)]
[Π (i : ι), normed_space 𝕜 (F' i)]
{Φ : E → Π (i : ι), F' i} :
cont_diff_at 𝕜 n Φ x ↔ ∀ (i : ι), cont_diff_at 𝕜 n (λ (x : E), Φ x i) x
theorem
cont_diff_pi
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{n : ℕ∞}
{ι : Type u_3}
[fintype ι]
{F' : ι → Type u_5}
[Π (i : ι), normed_add_comm_group (F' i)]
[Π (i : ι), normed_space 𝕜 (F' i)]
{Φ : E → Π (i : ι), F' i} :
theorem
cont_diff_apply
(𝕜 : Type u_1)
[nontrivially_normed_field 𝕜]
(E : Type uE)
[normed_add_comm_group E]
[normed_space 𝕜 E]
{n : ℕ∞}
{ι : Type u_3}
[fintype ι]
(i : ι) :
theorem
cont_diff_apply_apply
(𝕜 : Type u_1)
[nontrivially_normed_field 𝕜]
(E : Type uE)
[normed_add_comm_group E]
[normed_space 𝕜 E]
{n : ℕ∞}
{ι : Type u_3}
{ι' : Type u_4}
[fintype ι]
[fintype ι']
(i : ι)
(j : ι') :
Sum of two functions #
theorem
cont_diff_add
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞} :
theorem
cont_diff_within_at.add
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{x : E}
{n : ℕ∞}
{s : set E}
{f g : E → F}
(hf : cont_diff_within_at 𝕜 n f s x)
(hg : cont_diff_within_at 𝕜 n g s x) :
cont_diff_within_at 𝕜 n (λ (x : E), f x + g x) s x
The sum of two C^n functions within a set at a point is C^n within this set
at this point.
theorem
cont_diff_at.add
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{x : E}
{n : ℕ∞}
{f g : E → F}
(hf : cont_diff_at 𝕜 n f x)
(hg : cont_diff_at 𝕜 n g x) :
cont_diff_at 𝕜 n (λ (x : E), f x + g x) x
The sum of two C^n functions at a point is C^n at this point.
theorem
cont_diff.add
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{f g : E → F}
(hf : cont_diff 𝕜 n f)
(hg : cont_diff 𝕜 n g) :
The sum of two C^nfunctions is C^n.
theorem
cont_diff_on.add
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{s : set E}
{f g : E → F}
(hf : cont_diff_on 𝕜 n f s)
(hg : cont_diff_on 𝕜 n g s) :
cont_diff_on 𝕜 n (λ (x : E), f x + g x) s
The sum of two C^n functions on a domain is C^n.
theorem
iterated_fderiv_within_add_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{s : set E}
{x : E}
{i : ℕ}
{f g : E → F}
(hf : cont_diff_on 𝕜 ↑i f s)
(hg : cont_diff_on 𝕜 ↑i g s)
(hu : unique_diff_on 𝕜 s)
(hx : x ∈ s) :
iterated_fderiv_within 𝕜 i (f + g) s x = iterated_fderiv_within 𝕜 i f s x + iterated_fderiv_within 𝕜 i g s x
The iterated derivative of the sum of two functions is the sum of the iterated derivatives.
See also iterated_fderiv_within_add_apply', which uses the spelling (λ x, f x + g x)
instead of f + g.
theorem
iterated_fderiv_within_add_apply'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{s : set E}
{x : E}
{i : ℕ}
{f g : E → F}
(hf : cont_diff_on 𝕜 ↑i f s)
(hg : cont_diff_on 𝕜 ↑i g s)
(hu : unique_diff_on 𝕜 s)
(hx : x ∈ s) :
iterated_fderiv_within 𝕜 i (λ (x : E), f x + g x) s x = iterated_fderiv_within 𝕜 i f s x + iterated_fderiv_within 𝕜 i g s x
The iterated derivative of the sum of two functions is the sum of the iterated derivatives.
This is the same as iterated_fderiv_within_add_apply, but using the spelling (λ x, f x + g x)
instead of f + g, which can be handy for some rewrites.
TODO: use one form consistently.
theorem
iterated_fderiv_add_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{x : E}
{i : ℕ}
{f g : E → F}
(hf : cont_diff 𝕜 ↑i f)
(hg : cont_diff 𝕜 ↑i g) :
iterated_fderiv 𝕜 i (f + g) x = iterated_fderiv 𝕜 i f x + iterated_fderiv 𝕜 i g x
theorem
iterated_fderiv_add_apply'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{x : E}
{i : ℕ}
{f g : E → F}
(hf : cont_diff 𝕜 ↑i f)
(hg : cont_diff 𝕜 ↑i g) :
iterated_fderiv 𝕜 i (λ (x : E), f x + g x) x = iterated_fderiv 𝕜 i f x + iterated_fderiv 𝕜 i g x
Negative #
theorem
cont_diff_neg
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞} :
theorem
cont_diff_within_at.neg
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{x : E}
{n : ℕ∞}
{s : set E}
{f : E → F}
(hf : cont_diff_within_at 𝕜 n f s x) :
cont_diff_within_at 𝕜 n (λ (x : E), -f x) s x
The negative of a C^n function within a domain at a point is C^n within this domain at
this point.
theorem
cont_diff_at.neg
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{x : E}
{n : ℕ∞}
{f : E → F}
(hf : cont_diff_at 𝕜 n f x) :
cont_diff_at 𝕜 n (λ (x : E), -f x) x
The negative of a C^n function at a point is C^n at this point.
theorem
cont_diff.neg
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{f : E → F}
(hf : cont_diff 𝕜 n f) :
The negative of a C^nfunction is C^n.
theorem
cont_diff_on.neg
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{s : set E}
{f : E → F}
(hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (λ (x : E), -f x) s
The negative of a C^n function on a domain is C^n.
theorem
iterated_fderiv_within_neg_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{s : set E}
{x : E}
{i : ℕ}
{f : E → F}
(hu : unique_diff_on 𝕜 s)
(hx : x ∈ s) :
iterated_fderiv_within 𝕜 i (-f) s x = -iterated_fderiv_within 𝕜 i f s x
theorem
iterated_fderiv_neg_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{x : E}
{i : ℕ}
{f : E → F} :
iterated_fderiv 𝕜 i (-f) x = -iterated_fderiv 𝕜 i f x
Subtraction #
theorem
cont_diff_within_at.sub
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{x : E}
{n : ℕ∞}
{s : set E}
{f g : E → F}
(hf : cont_diff_within_at 𝕜 n f s x)
(hg : cont_diff_within_at 𝕜 n g s x) :
cont_diff_within_at 𝕜 n (λ (x : E), f x - g x) s x
The difference of two C^n functions within a set at a point is C^n within this set
at this point.
theorem
cont_diff_at.sub
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{x : E}
{n : ℕ∞}
{f g : E → F}
(hf : cont_diff_at 𝕜 n f x)
(hg : cont_diff_at 𝕜 n g x) :
cont_diff_at 𝕜 n (λ (x : E), f x - g x) x
The difference of two C^n functions at a point is C^n at this point.
theorem
cont_diff_on.sub
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{s : set E}
{f g : E → F}
(hf : cont_diff_on 𝕜 n f s)
(hg : cont_diff_on 𝕜 n g s) :
cont_diff_on 𝕜 n (λ (x : E), f x - g x) s
The difference of two C^n functions on a domain is C^n.
theorem
cont_diff.sub
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{f g : E → F}
(hf : cont_diff 𝕜 n f)
(hg : cont_diff 𝕜 n g) :
The difference of two C^n functions is C^n.
Sum of finitely many functions #
theorem
cont_diff_within_at.sum
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{ι : Type u_2}
{f : ι → E → F}
{s : finset ι}
{t : set E}
{x : E}
(h : ∀ (i : ι), i ∈ s → cont_diff_within_at 𝕜 n (λ (x : E), f i x) t x) :
cont_diff_within_at 𝕜 n (λ (x : E), s.sum (λ (i : ι), f i x)) t x
theorem
cont_diff_at.sum
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{ι : Type u_2}
{f : ι → E → F}
{s : finset ι}
{x : E}
(h : ∀ (i : ι), i ∈ s → cont_diff_at 𝕜 n (λ (x : E), f i x) x) :
cont_diff_at 𝕜 n (λ (x : E), s.sum (λ (i : ι), f i x)) x
theorem
cont_diff_on.sum
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{ι : Type u_2}
{f : ι → E → F}
{s : finset ι}
{t : set E}
(h : ∀ (i : ι), i ∈ s → cont_diff_on 𝕜 n (λ (x : E), f i x) t) :
cont_diff_on 𝕜 n (λ (x : E), s.sum (λ (i : ι), f i x)) t
theorem
cont_diff.sum
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{ι : Type u_2}
{f : ι → E → F}
{s : finset ι}
(h : ∀ (i : ι), i ∈ s → cont_diff 𝕜 n (λ (x : E), f i x)) :
Product of two functions #
theorem
cont_diff_mul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{n : ℕ∞}
{𝔸 : Type u_3}
[normed_ring 𝔸]
[normed_algebra 𝕜 𝔸] :
theorem
cont_diff_within_at.mul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{x : E}
{n : ℕ∞}
{𝔸 : Type u_3}
[normed_ring 𝔸]
[normed_algebra 𝕜 𝔸]
{s : set E}
{f g : E → 𝔸}
(hf : cont_diff_within_at 𝕜 n f s x)
(hg : cont_diff_within_at 𝕜 n g s x) :
cont_diff_within_at 𝕜 n (λ (x : E), f x * g x) s x
The product of two C^n functions within a set at a point is C^n within this set
at this point.
theorem
cont_diff_at.mul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{x : E}
{n : ℕ∞}
{𝔸 : Type u_3}
[normed_ring 𝔸]
[normed_algebra 𝕜 𝔸]
{f g : E → 𝔸}
(hf : cont_diff_at 𝕜 n f x)
(hg : cont_diff_at 𝕜 n g x) :
cont_diff_at 𝕜 n (λ (x : E), f x * g x) x
The product of two C^n functions at a point is C^n at this point.
theorem
cont_diff_on.mul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{n : ℕ∞}
{𝔸 : Type u_3}
[normed_ring 𝔸]
[normed_algebra 𝕜 𝔸]
{f g : E → 𝔸}
(hf : cont_diff_on 𝕜 n f s)
(hg : cont_diff_on 𝕜 n g s) :
cont_diff_on 𝕜 n (λ (x : E), f x * g x) s
The product of two C^n functions on a domain is C^n.
theorem
cont_diff.mul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{n : ℕ∞}
{𝔸 : Type u_3}
[normed_ring 𝔸]
[normed_algebra 𝕜 𝔸]
{f g : E → 𝔸}
(hf : cont_diff 𝕜 n f)
(hg : cont_diff 𝕜 n g) :
The product of two C^nfunctions is C^n.
theorem
cont_diff_within_at_prod'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{x : E}
{n : ℕ∞}
{𝔸' : Type u_4}
{ι : Type u_5}
[normed_comm_ring 𝔸']
[normed_algebra 𝕜 𝔸']
{t : finset ι}
{f : ι → E → 𝔸'}
(h : ∀ (i : ι), i ∈ t → cont_diff_within_at 𝕜 n (f i) s x) :
cont_diff_within_at 𝕜 n (t.prod (λ (i : ι), f i)) s x
theorem
cont_diff_within_at_prod
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{x : E}
{n : ℕ∞}
{𝔸' : Type u_4}
{ι : Type u_5}
[normed_comm_ring 𝔸']
[normed_algebra 𝕜 𝔸']
{t : finset ι}
{f : ι → E → 𝔸'}
(h : ∀ (i : ι), i ∈ t → cont_diff_within_at 𝕜 n (f i) s x) :
cont_diff_within_at 𝕜 n (λ (y : E), t.prod (λ (i : ι), f i y)) s x
theorem
cont_diff_at_prod'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{x : E}
{n : ℕ∞}
{𝔸' : Type u_4}
{ι : Type u_5}
[normed_comm_ring 𝔸']
[normed_algebra 𝕜 𝔸']
{t : finset ι}
{f : ι → E → 𝔸'}
(h : ∀ (i : ι), i ∈ t → cont_diff_at 𝕜 n (f i) x) :
cont_diff_at 𝕜 n (t.prod (λ (i : ι), f i)) x
theorem
cont_diff_at_prod
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{x : E}
{n : ℕ∞}
{𝔸' : Type u_4}
{ι : Type u_5}
[normed_comm_ring 𝔸']
[normed_algebra 𝕜 𝔸']
{t : finset ι}
{f : ι → E → 𝔸'}
(h : ∀ (i : ι), i ∈ t → cont_diff_at 𝕜 n (f i) x) :
cont_diff_at 𝕜 n (λ (y : E), t.prod (λ (i : ι), f i y)) x
theorem
cont_diff_on_prod'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{n : ℕ∞}
{𝔸' : Type u_4}
{ι : Type u_5}
[normed_comm_ring 𝔸']
[normed_algebra 𝕜 𝔸']
{t : finset ι}
{f : ι → E → 𝔸'}
(h : ∀ (i : ι), i ∈ t → cont_diff_on 𝕜 n (f i) s) :
cont_diff_on 𝕜 n (t.prod (λ (i : ι), f i)) s
theorem
cont_diff_on_prod
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{n : ℕ∞}
{𝔸' : Type u_4}
{ι : Type u_5}
[normed_comm_ring 𝔸']
[normed_algebra 𝕜 𝔸']
{t : finset ι}
{f : ι → E → 𝔸'}
(h : ∀ (i : ι), i ∈ t → cont_diff_on 𝕜 n (f i) s) :
cont_diff_on 𝕜 n (λ (y : E), t.prod (λ (i : ι), f i y)) s
theorem
cont_diff_prod'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{n : ℕ∞}
{𝔸' : Type u_4}
{ι : Type u_5}
[normed_comm_ring 𝔸']
[normed_algebra 𝕜 𝔸']
{t : finset ι}
{f : ι → E → 𝔸'}
(h : ∀ (i : ι), i ∈ t → cont_diff 𝕜 n (f i)) :
theorem
cont_diff_prod
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{n : ℕ∞}
{𝔸' : Type u_4}
{ι : Type u_5}
[normed_comm_ring 𝔸']
[normed_algebra 𝕜 𝔸']
{t : finset ι}
{f : ι → E → 𝔸'}
(h : ∀ (i : ι), i ∈ t → cont_diff 𝕜 n (f i)) :
theorem
cont_diff.pow
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{n : ℕ∞}
{𝔸 : Type u_3}
[normed_ring 𝔸]
[normed_algebra 𝕜 𝔸]
{f : E → 𝔸}
(hf : cont_diff 𝕜 n f)
(m : ℕ) :
theorem
cont_diff_within_at.pow
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{x : E}
{n : ℕ∞}
{𝔸 : Type u_3}
[normed_ring 𝔸]
[normed_algebra 𝕜 𝔸]
{f : E → 𝔸}
(hf : cont_diff_within_at 𝕜 n f s x)
(m : ℕ) :
cont_diff_within_at 𝕜 n (λ (y : E), f y ^ m) s x
theorem
cont_diff_at.pow
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{x : E}
{n : ℕ∞}
{𝔸 : Type u_3}
[normed_ring 𝔸]
[normed_algebra 𝕜 𝔸]
{f : E → 𝔸}
(hf : cont_diff_at 𝕜 n f x)
(m : ℕ) :
cont_diff_at 𝕜 n (λ (y : E), f y ^ m) x
theorem
cont_diff_on.pow
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{n : ℕ∞}
{𝔸 : Type u_3}
[normed_ring 𝔸]
[normed_algebra 𝕜 𝔸]
{f : E → 𝔸}
(hf : cont_diff_on 𝕜 n f s)
(m : ℕ) :
cont_diff_on 𝕜 n (λ (y : E), f y ^ m) s
theorem
cont_diff_within_at.div_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{x : E}
{𝕜' : Type u_6}
[normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
{f : E → 𝕜'}
{n : ℕ∞}
(hf : cont_diff_within_at 𝕜 n f s x)
(c : 𝕜') :
cont_diff_within_at 𝕜 n (λ (x : E), f x / c) s x
theorem
cont_diff_at.div_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{x : E}
{𝕜' : Type u_6}
[normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
{f : E → 𝕜'}
{n : ℕ∞}
(hf : cont_diff_at 𝕜 n f x)
(c : 𝕜') :
cont_diff_at 𝕜 n (λ (x : E), f x / c) x
theorem
cont_diff_on.div_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{𝕜' : Type u_6}
[normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
{f : E → 𝕜'}
{n : ℕ∞}
(hf : cont_diff_on 𝕜 n f s)
(c : 𝕜') :
cont_diff_on 𝕜 n (λ (x : E), f x / c) s
theorem
cont_diff.div_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{𝕜' : Type u_6}
[normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
{f : E → 𝕜'}
{n : ℕ∞}
(hf : cont_diff 𝕜 n f)
(c : 𝕜') :
Scalar multiplication #
theorem
cont_diff_smul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞} :
theorem
cont_diff_within_at.smul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{x : E}
{n : ℕ∞}
{s : set E}
{f : E → 𝕜}
{g : E → F}
(hf : cont_diff_within_at 𝕜 n f s x)
(hg : cont_diff_within_at 𝕜 n g s x) :
cont_diff_within_at 𝕜 n (λ (x : E), f x • g x) s x
The scalar multiplication of two C^n functions within a set at a point is C^n within this
set at this point.
theorem
cont_diff_at.smul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{x : E}
{n : ℕ∞}
{f : E → 𝕜}
{g : E → F}
(hf : cont_diff_at 𝕜 n f x)
(hg : cont_diff_at 𝕜 n g x) :
cont_diff_at 𝕜 n (λ (x : E), f x • g x) x
The scalar multiplication of two C^n functions at a point is C^n at this point.
theorem
cont_diff.smul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{f : E → 𝕜}
{g : E → F}
(hf : cont_diff 𝕜 n f)
(hg : cont_diff 𝕜 n g) :
The scalar multiplication of two C^n functions is C^n.
theorem
cont_diff_on.smul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{s : set E}
{f : E → 𝕜}
{g : E → F}
(hf : cont_diff_on 𝕜 n f s)
(hg : cont_diff_on 𝕜 n g s) :
cont_diff_on 𝕜 n (λ (x : E), f x • g x) s
The scalar multiplication of two C^n functions on a domain is C^n.
Constant scalar multiplication #
theorem
cont_diff_const_smul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{R : Type u_3}
[semiring R]
[module R F]
[smul_comm_class 𝕜 R F]
[has_continuous_const_smul R F]
(c : R) :
theorem
cont_diff_within_at.const_smul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{R : Type u_3}
[semiring R]
[module R F]
[smul_comm_class 𝕜 R F]
[has_continuous_const_smul R F]
{s : set E}
{f : E → F}
{x : E}
(c : R)
(hf : cont_diff_within_at 𝕜 n f s x) :
cont_diff_within_at 𝕜 n (λ (y : E), c • f y) s x
The scalar multiplication of a constant and a C^n function within a set at a point is C^n
within this set at this point.
theorem
cont_diff_at.const_smul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{R : Type u_3}
[semiring R]
[module R F]
[smul_comm_class 𝕜 R F]
[has_continuous_const_smul R F]
{f : E → F}
{x : E}
(c : R)
(hf : cont_diff_at 𝕜 n f x) :
cont_diff_at 𝕜 n (λ (y : E), c • f y) x
The scalar multiplication of a constant and a C^n function at a point is C^n at this
point.
theorem
cont_diff.const_smul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{R : Type u_3}
[semiring R]
[module R F]
[smul_comm_class 𝕜 R F]
[has_continuous_const_smul R F]
{f : E → F}
(c : R)
(hf : cont_diff 𝕜 n f) :
The scalar multiplication of a constant and a C^n function is C^n.
theorem
cont_diff_on.const_smul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{R : Type u_3}
[semiring R]
[module R F]
[smul_comm_class 𝕜 R F]
[has_continuous_const_smul R F]
{s : set E}
{f : E → F}
(c : R)
(hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (λ (y : E), c • f y) s
The scalar multiplication of a constant and a C^n on a domain is C^n.
theorem
iterated_fderiv_within_const_smul_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{s : set E}
{f : E → F}
{x : E}
{R : Type u_3}
[semiring R]
[module R F]
[smul_comm_class 𝕜 R F]
[has_continuous_const_smul R F]
{i : ℕ}
{a : R}
(hf : cont_diff_on 𝕜 ↑i f s)
(hu : unique_diff_on 𝕜 s)
(hx : x ∈ s) :
iterated_fderiv_within 𝕜 i (a • f) s x = a • iterated_fderiv_within 𝕜 i f s x
theorem
iterated_fderiv_const_smul_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{f : E → F}
{R : Type u_3}
[semiring R]
[module R F]
[smul_comm_class 𝕜 R F]
[has_continuous_const_smul R F]
{i : ℕ}
{a : R}
{x : E}
(hf : cont_diff 𝕜 ↑i f) :
iterated_fderiv 𝕜 i (a • f) x = a • iterated_fderiv 𝕜 i f x
Cartesian product of two functions #
theorem
cont_diff_within_at.prod_map'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{F' : Type u_4}
[normed_add_comm_group F']
[normed_space 𝕜 F']
{s : set E}
{t : set E'}
{f : E → F}
{g : E' → F'}
{p : E × E'}
(hf : cont_diff_within_at 𝕜 n f s p.fst)
(hg : cont_diff_within_at 𝕜 n g t p.snd) :
cont_diff_within_at 𝕜 n (prod.map f g) (s ×ˢ t) p
The product map of two C^n functions within a set at a point is C^n
within the product set at the product point.
theorem
cont_diff_within_at.prod_map
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{F' : Type u_4}
[normed_add_comm_group F']
[normed_space 𝕜 F']
{s : set E}
{t : set E'}
{f : E → F}
{g : E' → F'}
{x : E}
{y : E'}
(hf : cont_diff_within_at 𝕜 n f s x)
(hg : cont_diff_within_at 𝕜 n g t y) :
cont_diff_within_at 𝕜 n (prod.map f g) (s ×ˢ t) (x, y)
theorem
cont_diff_on.prod_map
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{E' : Type u_2}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space 𝕜 F']
{s : set E}
{t : set E'}
{f : E → F}
{g : E' → F'}
(hf : cont_diff_on 𝕜 n f s)
(hg : cont_diff_on 𝕜 n g t) :
cont_diff_on 𝕜 n (prod.map f g) (s ×ˢ t)
The product map of two C^n functions on a set is C^n on the product set.
theorem
cont_diff_at.prod_map
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{F' : Type u_4}
[normed_add_comm_group F']
[normed_space 𝕜 F']
{f : E → F}
{g : E' → F'}
{x : E}
{y : E'}
(hf : cont_diff_at 𝕜 n f x)
(hg : cont_diff_at 𝕜 n g y) :
cont_diff_at 𝕜 n (prod.map f g) (x, y)
The product map of two C^n functions within a set at a point is C^n
within the product set at the product point.
theorem
cont_diff_at.prod_map'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{F' : Type u_4}
[normed_add_comm_group F']
[normed_space 𝕜 F']
{f : E → F}
{g : E' → F'}
{p : E × E'}
(hf : cont_diff_at 𝕜 n f p.fst)
(hg : cont_diff_at 𝕜 n g p.snd) :
cont_diff_at 𝕜 n (prod.map f g) p
The product map of two C^n functions within a set at a point is C^n
within the product set at the product point.
theorem
cont_diff.prod_map
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{F' : Type u_4}
[normed_add_comm_group F']
[normed_space 𝕜 F']
{f : E → F}
{g : E' → F'}
(hf : cont_diff 𝕜 n f)
(hg : cont_diff 𝕜 n g) :
The product map of two C^n functions is C^n.
theorem
cont_diff_prod_mk_left
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
(f₀ : F) :
cont_diff 𝕜 n (λ (e : E), (e, f₀))
theorem
cont_diff_prod_mk_right
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
(e₀ : E) :
cont_diff 𝕜 n (λ (f : F), (e₀, f))
Inversion in a complete normed algebra #
theorem
cont_diff_at_ring_inverse
(𝕜 : Type u_1)
[nontrivially_normed_field 𝕜]
{n : ℕ∞}
{R : Type u_3}
[normed_ring R]
[normed_algebra 𝕜 R]
[complete_space R]
(x : Rˣ) :
cont_diff_at 𝕜 n ring.inverse ↑x
In a complete normed algebra, the operation of inversion is C^n, for all n, at each
invertible element. The proof is by induction, bootstrapping using an identity expressing the
derivative of inversion as a bilinear map of inversion itself.
theorem
cont_diff_at_inv
(𝕜 : Type u_1)
[nontrivially_normed_field 𝕜]
{𝕜' : Type u_4}
[normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
[complete_space 𝕜']
{x : 𝕜'}
(hx : x ≠ 0)
{n : ℕ∞} :
cont_diff_at 𝕜 n has_inv.inv x
theorem
cont_diff_on_inv
(𝕜 : Type u_1)
[nontrivially_normed_field 𝕜]
{𝕜' : Type u_4}
[normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
[complete_space 𝕜']
{n : ℕ∞} :
cont_diff_on 𝕜 n has_inv.inv {0}ᶜ
theorem
cont_diff_within_at.inv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{x : E}
{𝕜' : Type u_4}
[normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
[complete_space 𝕜']
{f : E → 𝕜'}
{n : ℕ∞}
(hf : cont_diff_within_at 𝕜 n f s x)
(hx : f x ≠ 0) :
cont_diff_within_at 𝕜 n (λ (x : E), (f x)⁻¹) s x
theorem
cont_diff_on.inv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{𝕜' : Type u_4}
[normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
[complete_space 𝕜']
{f : E → 𝕜'}
{n : ℕ∞}
(hf : cont_diff_on 𝕜 n f s)
(h : ∀ (x : E), x ∈ s → f x ≠ 0) :
cont_diff_on 𝕜 n (λ (x : E), (f x)⁻¹) s
theorem
cont_diff_at.inv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{x : E}
{𝕜' : Type u_4}
[normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
[complete_space 𝕜']
{f : E → 𝕜'}
{n : ℕ∞}
(hf : cont_diff_at 𝕜 n f x)
(hx : f x ≠ 0) :
cont_diff_at 𝕜 n (λ (x : E), (f x)⁻¹) x
theorem
cont_diff.inv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{𝕜' : Type u_4}
[normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
[complete_space 𝕜']
{f : E → 𝕜'}
{n : ℕ∞}
(hf : cont_diff 𝕜 n f)
(h : ∀ (x : E), f x ≠ 0) :
theorem
cont_diff_within_at.div
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{x : E}
[complete_space 𝕜]
{f g : E → 𝕜}
{n : ℕ∞}
(hf : cont_diff_within_at 𝕜 n f s x)
(hg : cont_diff_within_at 𝕜 n g s x)
(hx : g x ≠ 0) :
cont_diff_within_at 𝕜 n (λ (x : E), f x / g x) s x
theorem
cont_diff_on.div
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
[complete_space 𝕜]
{f g : E → 𝕜}
{n : ℕ∞}
(hf : cont_diff_on 𝕜 n f s)
(hg : cont_diff_on 𝕜 n g s)
(h₀ : ∀ (x : E), x ∈ s → g x ≠ 0) :
cont_diff_on 𝕜 n (f / g) s
theorem
cont_diff_at.div
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{x : E}
[complete_space 𝕜]
{f g : E → 𝕜}
{n : ℕ∞}
(hf : cont_diff_at 𝕜 n f x)
(hg : cont_diff_at 𝕜 n g x)
(hx : g x ≠ 0) :
cont_diff_at 𝕜 n (λ (x : E), f x / g x) x
theorem
cont_diff.div
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
[complete_space 𝕜]
{f g : E → 𝕜}
{n : ℕ∞}
(hf : cont_diff 𝕜 n f)
(hg : cont_diff 𝕜 n g)
(h0 : ∀ (x : E), g x ≠ 0) :
Inversion of continuous linear maps between Banach spaces #
theorem
cont_diff_at_map_inverse
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
[complete_space E]
(e : E ≃L[𝕜] F) :
At a continuous linear equivalence e : E ≃L[𝕜] F between Banach spaces, the operation of
inversion is C^n, for all n.
theorem
local_homeomorph.cont_diff_at_symm
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
[complete_space E]
(f : local_homeomorph E F)
{f₀' : E ≃L[𝕜] F}
{a : F}
(ha : a ∈ f.to_local_equiv.target)
(hf₀' : has_fderiv_at ⇑f ↑f₀' (⇑(f.symm) a))
(hf : cont_diff_at 𝕜 n ⇑f (⇑(f.symm) a)) :
cont_diff_at 𝕜 n ⇑(f.symm) a
If f is a local homeomorphism and the point a is in its target,
and if f is n times continuously differentiable at f.symm a,
and if the derivative at f.symm a is a continuous linear equivalence,
then f.symm is n times continuously differentiable at the point a.
This is one of the easy parts of the inverse function theorem: it assumes that we already have an inverse function.
theorem
homeomorph.cont_diff_symm
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
[complete_space E]
(f : E ≃ₜ F)
{f₀' : E → (E ≃L[𝕜] F)}
(hf₀' : ∀ (a : E), has_fderiv_at ⇑f ↑(f₀' a) a)
(hf : cont_diff 𝕜 n ⇑f) :
If f is an n times continuously differentiable homeomorphism,
and if the derivative of f at each point is a continuous linear equivalence,
then f.symm is n times continuously differentiable.
This is one of the easy parts of the inverse function theorem: it assumes that we already have an inverse function.
theorem
local_homeomorph.cont_diff_at_symm_deriv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{n : ℕ∞}
[complete_space 𝕜]
(f : local_homeomorph 𝕜 𝕜)
{f₀' a : 𝕜}
(h₀ : f₀' ≠ 0)
(ha : a ∈ f.to_local_equiv.target)
(hf₀' : has_deriv_at ⇑f f₀' (⇑(f.symm) a))
(hf : cont_diff_at 𝕜 n ⇑f (⇑(f.symm) a)) :
cont_diff_at 𝕜 n ⇑(f.symm) a
Let f be a local homeomorphism of a nontrivially normed field, let a be a point in its
target. if f is n times continuously differentiable at f.symm a, and if the derivative at
f.symm a is nonzero, then f.symm is n times continuously differentiable at the point a.
This is one of the easy parts of the inverse function theorem: it assumes that we already have an inverse function.
theorem
homeomorph.cont_diff_symm_deriv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{n : ℕ∞}
[complete_space 𝕜]
(f : 𝕜 ≃ₜ 𝕜)
{f' : 𝕜 → 𝕜}
(h₀ : ∀ (x : 𝕜), f' x ≠ 0)
(hf' : ∀ (x : 𝕜), has_deriv_at ⇑f (f' x) x)
(hf : cont_diff 𝕜 n ⇑f) :
Let f be an n times continuously differentiable homeomorphism of a nontrivially normed
field. Suppose that the derivative of f is never equal to zero. Then f.symm is n times
continuously differentiable.
This is one of the easy parts of the inverse function theorem: it assumes that we already have an inverse function.
Finite dimensional results #
theorem
cont_diff_on_clm_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
[complete_space 𝕜]
{n : ℕ∞}
{f : E → (F →L[𝕜] G)}
{s : set E}
[finite_dimensional 𝕜 F] :
cont_diff_on 𝕜 n f s ↔ ∀ (y : F), cont_diff_on 𝕜 n (λ (x : E), ⇑(f x) y) s
A family of continuous linear maps is C^n on s if all its applications are.
theorem
cont_diff_clm_apply_iff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
[complete_space 𝕜]
{n : ℕ∞}
{f : E → (F →L[𝕜] G)}
[finite_dimensional 𝕜 F] :
theorem
cont_diff_succ_iff_fderiv_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
[complete_space 𝕜]
[finite_dimensional 𝕜 E]
{n : ℕ}
{f : E → F} :
This is a useful lemma to prove that a certain operation preserves functions being C^n.
When you do induction on n, this gives a useful characterization of a function being C^(n+1),
assuming you have already computed the derivative. The advantage of this version over
cont_diff_succ_iff_fderiv is that both occurences of cont_diff are for functions with the same
domain and codomain (E and F). This is not the case for cont_diff_succ_iff_fderiv, which
often requires an inconvenient need to generalize F, which results in universe issues
(see the discussion in the section of cont_diff.comp).
This lemma avoids these universe issues, but only applies for finite dimensional E.
theorem
cont_diff_on_succ_of_fderiv_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
[complete_space 𝕜]
[finite_dimensional 𝕜 E]
{n : ℕ}
{f : E → F}
{s : set E}
(hf : differentiable_on 𝕜 f s)
(h : ∀ (y : E), cont_diff_on 𝕜 ↑n (λ (x : E), ⇑(fderiv_within 𝕜 f s x) y) s) :
cont_diff_on 𝕜 ↑(n + 1) f s
theorem
cont_diff_on_succ_iff_fderiv_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
[complete_space 𝕜]
[finite_dimensional 𝕜 E]
{n : ℕ}
{f : E → F}
{s : set E}
(hs : unique_diff_on 𝕜 s) :
cont_diff_on 𝕜 ↑(n + 1) f s ↔ differentiable_on 𝕜 f s ∧ ∀ (y : E), cont_diff_on 𝕜 ↑n (λ (x : E), ⇑(fderiv_within 𝕜 f s x) y) s
Results over ℝ or ℂ #
The results in this section rely on the Mean Value Theorem, and therefore hold only over ℝ (and
its extension fields such as ℂ).
theorem
has_ftaylor_series_up_to_on.has_strict_fderiv_at
{n : ℕ∞}
{𝕂 : Type u_3}
[is_R_or_C 𝕂]
{E' : Type u_4}
[normed_add_comm_group E']
[normed_space 𝕂 E']
{F' : Type u_5}
[normed_add_comm_group F']
[normed_space 𝕂 F']
{s : set E'}
{f : E' → F'}
{x : E'}
{p : E' → formal_multilinear_series 𝕂 E' F'}
(hf : has_ftaylor_series_up_to_on n f p s)
(hn : 1 ≤ n)
(hs : s ∈ nhds x) :
has_strict_fderiv_at f (⇑(continuous_multilinear_curry_fin1 𝕂 E' F') (p x 1)) x
If a function has a Taylor series at order at least 1, then at points in the interior of the
domain of definition, the term of order 1 of this series is a strict derivative of f.
theorem
cont_diff_at.has_strict_fderiv_at'
{n : ℕ∞}
{𝕂 : Type u_3}
[is_R_or_C 𝕂]
{E' : Type u_4}
[normed_add_comm_group E']
[normed_space 𝕂 E']
{F' : Type u_5}
[normed_add_comm_group F']
[normed_space 𝕂 F']
{f : E' → F'}
{f' : E' →L[𝕂] F'}
{x : E'}
(hf : cont_diff_at 𝕂 n f x)
(hf' : has_fderiv_at f f' x)
(hn : 1 ≤ n) :
has_strict_fderiv_at f f' x
If a function is C^n with 1 ≤ n around a point, and its derivative at that point is given to
us as f', then f' is also a strict derivative.
theorem
cont_diff_at.has_strict_deriv_at'
{n : ℕ∞}
{𝕂 : Type u_3}
[is_R_or_C 𝕂]
{F' : Type u_5}
[normed_add_comm_group F']
[normed_space 𝕂 F']
{f : 𝕂 → F'}
{f' : F'}
{x : 𝕂}
(hf : cont_diff_at 𝕂 n f x)
(hf' : has_deriv_at f f' x)
(hn : 1 ≤ n) :
has_strict_deriv_at f f' x
If a function is C^n with 1 ≤ n around a point, and its derivative at that point is given to
us as f', then f' is also a strict derivative.
theorem
cont_diff_at.has_strict_fderiv_at
{n : ℕ∞}
{𝕂 : Type u_3}
[is_R_or_C 𝕂]
{E' : Type u_4}
[normed_add_comm_group E']
[normed_space 𝕂 E']
{F' : Type u_5}
[normed_add_comm_group F']
[normed_space 𝕂 F']
{f : E' → F'}
{x : E'}
(hf : cont_diff_at 𝕂 n f x)
(hn : 1 ≤ n) :
has_strict_fderiv_at f (fderiv 𝕂 f x) x
If a function is C^n with 1 ≤ n around a point, then the derivative of f at this point
is also a strict derivative.
theorem
cont_diff_at.has_strict_deriv_at
{n : ℕ∞}
{𝕂 : Type u_3}
[is_R_or_C 𝕂]
{F' : Type u_5}
[normed_add_comm_group F']
[normed_space 𝕂 F']
{f : 𝕂 → F'}
{x : 𝕂}
(hf : cont_diff_at 𝕂 n f x)
(hn : 1 ≤ n) :
has_strict_deriv_at f (deriv f x) x
If a function is C^n with 1 ≤ n around a point, then the derivative of f at this point
is also a strict derivative.
theorem
cont_diff.has_strict_fderiv_at
{n : ℕ∞}
{𝕂 : Type u_3}
[is_R_or_C 𝕂]
{E' : Type u_4}
[normed_add_comm_group E']
[normed_space 𝕂 E']
{F' : Type u_5}
[normed_add_comm_group F']
[normed_space 𝕂 F']
{f : E' → F'}
{x : E'}
(hf : cont_diff 𝕂 n f)
(hn : 1 ≤ n) :
has_strict_fderiv_at f (fderiv 𝕂 f x) x
If a function is C^n with 1 ≤ n, then the derivative of f is also a strict derivative.
theorem
cont_diff.has_strict_deriv_at
{n : ℕ∞}
{𝕂 : Type u_3}
[is_R_or_C 𝕂]
{F' : Type u_5}
[normed_add_comm_group F']
[normed_space 𝕂 F']
{f : 𝕂 → F'}
{x : 𝕂}
(hf : cont_diff 𝕂 n f)
(hn : 1 ≤ n) :
has_strict_deriv_at f (deriv f x) x
If a function is C^n with 1 ≤ n, then the derivative of f is also a strict derivative.
theorem
has_ftaylor_series_up_to_on.exists_lipschitz_on_with_of_nnnorm_lt
{E : Type u_1}
{F : Type u_2}
[normed_add_comm_group E]
[normed_space ℝ E]
[normed_add_comm_group F]
[normed_space ℝ F]
{f : E → F}
{p : E → formal_multilinear_series ℝ E F}
{s : set E}
{x : E}
(hf : has_ftaylor_series_up_to_on 1 f p (has_insert.insert x s))
(hs : convex ℝ s)
(K : nnreal)
(hK : ‖p x 1‖₊ < K) :
∃ (t : set E) (H : t ∈ nhds_within x s), lipschitz_on_with K f t
If f has a formal Taylor series p up to order 1 on {x} ∪ s, where s is a convex set,
and ‖p x 1‖₊ < K, then f is K-Lipschitz in a neighborhood of x within s.
theorem
has_ftaylor_series_up_to_on.exists_lipschitz_on_with
{E : Type u_1}
{F : Type u_2}
[normed_add_comm_group E]
[normed_space ℝ E]
[normed_add_comm_group F]
[normed_space ℝ F]
{f : E → F}
{p : E → formal_multilinear_series ℝ E F}
{s : set E}
{x : E}
(hf : has_ftaylor_series_up_to_on 1 f p (has_insert.insert x s))
(hs : convex ℝ s) :
∃ (K : nnreal) (t : set E) (H : t ∈ nhds_within x s), lipschitz_on_with K f t
If f has a formal Taylor series p up to order 1 on {x} ∪ s, where s is a convex set,
then f is Lipschitz in a neighborhood of x within s.
theorem
cont_diff_within_at.exists_lipschitz_on_with
{E : Type u_1}
{F : Type u_2}
[normed_add_comm_group E]
[normed_space ℝ E]
[normed_add_comm_group F]
[normed_space ℝ F]
{f : E → F}
{s : set E}
{x : E}
(hf : cont_diff_within_at ℝ 1 f s x)
(hs : convex ℝ s) :
∃ (K : nnreal) (t : set E) (H : t ∈ nhds_within x s), lipschitz_on_with K f t
If f is C^1 within a conves set s at x, then it is Lipschitz on a neighborhood of x
within s.
theorem
cont_diff_at.exists_lipschitz_on_with_of_nnnorm_lt
{𝕂 : Type u_3}
[is_R_or_C 𝕂]
{E' : Type u_4}
[normed_add_comm_group E']
[normed_space 𝕂 E']
{F' : Type u_5}
[normed_add_comm_group F']
[normed_space 𝕂 F']
{f : E' → F'}
{x : E'}
(hf : cont_diff_at 𝕂 1 f x)
(K : nnreal)
(hK : ‖fderiv 𝕂 f x‖₊ < K) :
If f is C^1 at x and K > ‖fderiv 𝕂 f x‖, then f is K-Lipschitz in a neighborhood of
x.
theorem
cont_diff_at.exists_lipschitz_on_with
{𝕂 : Type u_3}
[is_R_or_C 𝕂]
{E' : Type u_4}
[normed_add_comm_group E']
[normed_space 𝕂 E']
{F' : Type u_5}
[normed_add_comm_group F']
[normed_space 𝕂 F']
{f : E' → F'}
{x : E'}
(hf : cont_diff_at 𝕂 1 f x) :
If f is C^1 at x, then f is Lipschitz in a neighborhood of x.
One dimension #
All results up to now have been expressed in terms of the general Fréchet derivative fderiv. For
maps defined on the field, the one-dimensional derivative deriv is often easier to use. In this
paragraph, we reformulate some higher smoothness results in terms of deriv.
theorem
cont_diff_on_succ_iff_deriv_within
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{f₂ : 𝕜 → F}
{s₂ : set 𝕜}
{n : ℕ}
(hs : unique_diff_on 𝕜 s₂) :
cont_diff_on 𝕜 ↑(n + 1) f₂ s₂ ↔ differentiable_on 𝕜 f₂ s₂ ∧ cont_diff_on 𝕜 ↑n (deriv_within f₂ s₂) s₂
A function is C^(n + 1) on a domain with unique derivatives if and only if it is
differentiable there, and its derivative (formulated with deriv_within) is C^n.
theorem
cont_diff_on_succ_iff_deriv_of_open
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{f₂ : 𝕜 → F}
{s₂ : set 𝕜}
{n : ℕ}
(hs : is_open s₂) :
cont_diff_on 𝕜 ↑(n + 1) f₂ s₂ ↔ differentiable_on 𝕜 f₂ s₂ ∧ cont_diff_on 𝕜 ↑n (deriv f₂) s₂
A function is C^(n + 1) on an open domain if and only if it is
differentiable there, and its derivative (formulated with deriv) is C^n.
theorem
cont_diff_on_top_iff_deriv_within
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{f₂ : 𝕜 → F}
{s₂ : set 𝕜}
(hs : unique_diff_on 𝕜 s₂) :
cont_diff_on 𝕜 ⊤ f₂ s₂ ↔ differentiable_on 𝕜 f₂ s₂ ∧ cont_diff_on 𝕜 ⊤ (deriv_within f₂ s₂) s₂
A function is C^∞ on a domain with unique derivatives if and only if it is differentiable
there, and its derivative (formulated with deriv_within) is C^∞.
theorem
cont_diff_on_top_iff_deriv_of_open
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{f₂ : 𝕜 → F}
{s₂ : set 𝕜}
(hs : is_open s₂) :
cont_diff_on 𝕜 ⊤ f₂ s₂ ↔ differentiable_on 𝕜 f₂ s₂ ∧ cont_diff_on 𝕜 ⊤ (deriv f₂) s₂
A function is C^∞ on an open domain if and only if it is differentiable
there, and its derivative (formulated with deriv) is C^∞.
theorem
cont_diff_on.deriv_within
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{m n : ℕ∞}
{f₂ : 𝕜 → F}
{s₂ : set 𝕜}
(hf : cont_diff_on 𝕜 n f₂ s₂)
(hs : unique_diff_on 𝕜 s₂)
(hmn : m + 1 ≤ n) :
cont_diff_on 𝕜 m (deriv_within f₂ s₂) s₂
theorem
cont_diff_on.deriv_of_open
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{m n : ℕ∞}
{f₂ : 𝕜 → F}
{s₂ : set 𝕜}
(hf : cont_diff_on 𝕜 n f₂ s₂)
(hs : is_open s₂)
(hmn : m + 1 ≤ n) :
cont_diff_on 𝕜 m (deriv f₂) s₂
theorem
cont_diff_on.continuous_on_deriv_within
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{f₂ : 𝕜 → F}
{s₂ : set 𝕜}
(h : cont_diff_on 𝕜 n f₂ s₂)
(hs : unique_diff_on 𝕜 s₂)
(hn : 1 ≤ n) :
continuous_on (deriv_within f₂ s₂) s₂
theorem
cont_diff_on.continuous_on_deriv_of_open
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{f₂ : 𝕜 → F}
{s₂ : set 𝕜}
(h : cont_diff_on 𝕜 n f₂ s₂)
(hs : is_open s₂)
(hn : 1 ≤ n) :
continuous_on (deriv f₂) s₂
theorem
cont_diff_succ_iff_deriv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{f₂ : 𝕜 → F}
{n : ℕ} :
A function is C^(n + 1) if and only if it is differentiable,
and its derivative (formulated in terms of deriv) is C^n.
theorem
cont_diff_one_iff_deriv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{f₂ : 𝕜 → F} :
cont_diff 𝕜 1 f₂ ↔ differentiable 𝕜 f₂ ∧ continuous (deriv f₂)
theorem
cont_diff_top_iff_deriv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{f₂ : 𝕜 → F} :
A function is C^∞ if and only if it is differentiable,
and its derivative (formulated in terms of deriv) is C^∞.
theorem
cont_diff.continuous_deriv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{n : ℕ∞}
{f₂ : 𝕜 → F}
(h : cont_diff 𝕜 n f₂)
(hn : 1 ≤ n) :
continuous (deriv f₂)
theorem
cont_diff.iterate_deriv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
(n : ℕ)
{f₂ : 𝕜 → F}
(hf : cont_diff 𝕜 ⊤ f₂) :
theorem
cont_diff.iterate_deriv'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
(n k : ℕ)
{f₂ : 𝕜 → F}
(hf : cont_diff 𝕜 ↑(n + k) f₂) :
Restricting from ℂ to ℝ, or generally from 𝕜' to 𝕜 #
If a function is n times continuously differentiable over ℂ, then it is n times continuously
differentiable over ℝ. In this paragraph, we give variants of this statement, in the general
situation where ℂ and ℝ are replaced respectively by 𝕜' and 𝕜 where 𝕜' is a normed algebra
over 𝕜.
theorem
has_ftaylor_series_up_to_on.restrict_scalars
(𝕜 : Type u_1)
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{s : set E}
{f : E → F}
{n : ℕ∞}
{𝕜' : Type u_3}
[nontrivially_normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
[normed_space 𝕜' E]
[is_scalar_tower 𝕜 𝕜' E]
[normed_space 𝕜' F]
[is_scalar_tower 𝕜 𝕜' F]
{p' : E → formal_multilinear_series 𝕜' E F}
(h : has_ftaylor_series_up_to_on n f p' s) :
has_ftaylor_series_up_to_on n f (λ (x : E), formal_multilinear_series.restrict_scalars 𝕜 (p' x)) s
theorem
cont_diff_within_at.restrict_scalars
(𝕜 : Type u_1)
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{s : set E}
{f : E → F}
{x : E}
{n : ℕ∞}
{𝕜' : Type u_3}
[nontrivially_normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
[normed_space 𝕜' E]
[is_scalar_tower 𝕜 𝕜' E]
[normed_space 𝕜' F]
[is_scalar_tower 𝕜 𝕜' F]
(h : cont_diff_within_at 𝕜' n f s x) :
cont_diff_within_at 𝕜 n f s x
theorem
cont_diff_on.restrict_scalars
(𝕜 : Type u_1)
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{s : set E}
{f : E → F}
{n : ℕ∞}
{𝕜' : Type u_3}
[nontrivially_normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
[normed_space 𝕜' E]
[is_scalar_tower 𝕜 𝕜' E]
[normed_space 𝕜' F]
[is_scalar_tower 𝕜 𝕜' F]
(h : cont_diff_on 𝕜' n f s) :
cont_diff_on 𝕜 n f s
theorem
cont_diff_at.restrict_scalars
(𝕜 : Type u_1)
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{f : E → F}
{x : E}
{n : ℕ∞}
{𝕜' : Type u_3}
[nontrivially_normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
[normed_space 𝕜' E]
[is_scalar_tower 𝕜 𝕜' E]
[normed_space 𝕜' F]
[is_scalar_tower 𝕜 𝕜' F]
(h : cont_diff_at 𝕜' n f x) :
cont_diff_at 𝕜 n f x
theorem
cont_diff.restrict_scalars
(𝕜 : Type u_1)
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{f : E → F}
{n : ℕ∞}
{𝕜' : Type u_3}
[nontrivially_normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
[normed_space 𝕜' E]
[is_scalar_tower 𝕜 𝕜' E]
[normed_space 𝕜' F]
[is_scalar_tower 𝕜 𝕜' F]
(h : cont_diff 𝕜' n f) :
cont_diff 𝕜 n f
Quantitative bounds #
theorem
continuous_linear_map.norm_iterated_fderiv_within_le_of_bilinear_aux
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{Du Eu Fu Gu : Type u}
[normed_add_comm_group Du]
[normed_space 𝕜 Du]
[normed_add_comm_group Eu]
[normed_space 𝕜 Eu]
[normed_add_comm_group Fu]
[normed_space 𝕜 Fu]
[normed_add_comm_group Gu]
[normed_space 𝕜 Gu]
(B : Eu →L[𝕜] Fu →L[𝕜] Gu)
{f : Du → Eu}
{g : Du → Fu}
{n : ℕ}
{s : set Du}
{x : Du}
(hf : cont_diff_on 𝕜 ↑n f s)
(hg : cont_diff_on 𝕜 ↑n g s)
(hs : unique_diff_on 𝕜 s)
(hx : x ∈ s) :
Bounding the norm of the iterated derivative of B (f x) (g x) within a set in terms of the
iterated derivatives of f and g when B is bilinear. This lemma is an auxiliary version
assuming all spaces live in the same universe, to enable an induction. Use instead
continuous_linear_map.norm_iterated_fderiv_within_le_of_bilinear that removes this assumption.
theorem
continuous_linear_map.norm_iterated_fderiv_within_le_of_bilinear
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{D : Type uD}
[normed_add_comm_group D]
[normed_space 𝕜 D]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
(B : E →L[𝕜] F →L[𝕜] G)
{f : D → E}
{g : D → F}
{N : ℕ∞}
{s : set D}
{x : D}
(hf : cont_diff_on 𝕜 N f s)
(hg : cont_diff_on 𝕜 N g s)
(hs : unique_diff_on 𝕜 s)
(hx : x ∈ s)
{n : ℕ}
(hn : ↑n ≤ N) :
Bounding the norm of the iterated derivative of B (f x) (g x) within a set in terms of the
iterated derivatives of f and g when B is bilinear:
‖D^n (x ↦ B (f x) (g x))‖ ≤ ‖B‖ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖
theorem
continuous_linear_map.norm_iterated_fderiv_le_of_bilinear
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{D : Type uD}
[normed_add_comm_group D]
[normed_space 𝕜 D]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
(B : E →L[𝕜] F →L[𝕜] G)
{f : D → E}
{g : D → F}
{N : ℕ∞}
(hf : cont_diff 𝕜 N f)
(hg : cont_diff 𝕜 N g)
(x : D)
{n : ℕ}
(hn : ↑n ≤ N) :
Bounding the norm of the iterated derivative of B (f x) (g x) in terms of the
iterated derivatives of f and g when B is bilinear:
‖D^n (x ↦ B (f x) (g x))‖ ≤ ‖B‖ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖
theorem
continuous_linear_map.norm_iterated_fderiv_within_le_of_bilinear_of_le_one
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{D : Type uD}
[normed_add_comm_group D]
[normed_space 𝕜 D]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
(B : E →L[𝕜] F →L[𝕜] G)
{f : D → E}
{g : D → F}
{N : ℕ∞}
{s : set D}
{x : D}
(hf : cont_diff_on 𝕜 N f s)
(hg : cont_diff_on 𝕜 N g s)
(hs : unique_diff_on 𝕜 s)
(hx : x ∈ s)
{n : ℕ}
(hn : ↑n ≤ N)
(hB : ‖B‖ ≤ 1) :
Bounding the norm of the iterated derivative of B (f x) (g x) within a set in terms of the
iterated derivatives of f and g when B is bilinear of norm at most 1:
‖D^n (x ↦ B (f x) (g x))‖ ≤ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖
theorem
continuous_linear_map.norm_iterated_fderiv_le_of_bilinear_of_le_one
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{D : Type uD}
[normed_add_comm_group D]
[normed_space 𝕜 D]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
(B : E →L[𝕜] F →L[𝕜] G)
{f : D → E}
{g : D → F}
{N : ℕ∞}
(hf : cont_diff 𝕜 N f)
(hg : cont_diff 𝕜 N g)
(x : D)
{n : ℕ}
(hn : ↑n ≤ N)
(hB : ‖B‖ ≤ 1) :
Bounding the norm of the iterated derivative of B (f x) (g x) in terms of the
iterated derivatives of f and g when B is bilinear of norm at most 1:
‖D^n (x ↦ B (f x) (g x))‖ ≤ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖
theorem
norm_iterated_fderiv_within_smul_le
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{s : set E}
{𝕜' : Type u_3}
[normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
[normed_space 𝕜' F]
[is_scalar_tower 𝕜 𝕜' F]
{f : E → 𝕜'}
{g : E → F}
{N : ℕ∞}
(hf : cont_diff_on 𝕜 N f s)
(hg : cont_diff_on 𝕜 N g s)
(hs : unique_diff_on 𝕜 s)
{x : E}
(hx : x ∈ s)
{n : ℕ}
(hn : ↑n ≤ N) :
‖iterated_fderiv_within 𝕜 n (λ (y : E), f y • g y) s x‖ ≤ (finset.range (n + 1)).sum (λ (i : ℕ), ↑(n.choose i) * ‖iterated_fderiv_within 𝕜 i f s x‖ * ‖iterated_fderiv_within 𝕜 (n - i) g s x‖)
theorem
norm_iterated_fderiv_smul_le
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{𝕜' : Type u_3}
[normed_field 𝕜']
[normed_algebra 𝕜 𝕜']
[normed_space 𝕜' F]
[is_scalar_tower 𝕜 𝕜' F]
{f : E → 𝕜'}
{g : E → F}
{N : ℕ∞}
(hf : cont_diff 𝕜 N f)
(hg : cont_diff 𝕜 N g)
(x : E)
{n : ℕ}
(hn : ↑n ≤ N) :
‖iterated_fderiv 𝕜 n (λ (y : E), f y • g y) x‖ ≤ (finset.range (n + 1)).sum (λ (i : ℕ), ↑(n.choose i) * ‖iterated_fderiv 𝕜 i f x‖ * ‖iterated_fderiv 𝕜 (n - i) g x‖)
theorem
norm_iterated_fderiv_within_mul_le
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{A : Type u_3}
[normed_ring A]
[normed_algebra 𝕜 A]
{f g : E → A}
{N : ℕ∞}
(hf : cont_diff_on 𝕜 N f s)
(hg : cont_diff_on 𝕜 N g s)
(hs : unique_diff_on 𝕜 s)
{x : E}
(hx : x ∈ s)
{n : ℕ}
(hn : ↑n ≤ N) :
‖iterated_fderiv_within 𝕜 n (λ (y : E), f y * g y) s x‖ ≤ (finset.range (n + 1)).sum (λ (i : ℕ), ↑(n.choose i) * ‖iterated_fderiv_within 𝕜 i f s x‖ * ‖iterated_fderiv_within 𝕜 (n - i) g s x‖)
theorem
norm_iterated_fderiv_mul_le
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{A : Type u_3}
[normed_ring A]
[normed_algebra 𝕜 A]
{f g : E → A}
{N : ℕ∞}
(hf : cont_diff 𝕜 N f)
(hg : cont_diff 𝕜 N g)
(x : E)
{n : ℕ}
(hn : ↑n ≤ N) :
‖iterated_fderiv 𝕜 n (λ (y : E), f y * g y) x‖ ≤ (finset.range (n + 1)).sum (λ (i : ℕ), ↑(n.choose i) * ‖iterated_fderiv 𝕜 i f x‖ * ‖iterated_fderiv 𝕜 (n - i) g x‖)
theorem
norm_iterated_fderiv_within_comp_le_aux
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{Fu Gu : Type u}
[normed_add_comm_group Fu]
[normed_space 𝕜 Fu]
[normed_add_comm_group Gu]
[normed_space 𝕜 Gu]
{g : Fu → Gu}
{f : E → Fu}
{n : ℕ}
{s : set E}
{t : set Fu}
{x : E}
(hg : cont_diff_on 𝕜 ↑n g t)
(hf : cont_diff_on 𝕜 ↑n f s)
(ht : unique_diff_on 𝕜 t)
(hs : unique_diff_on 𝕜 s)
(hst : set.maps_to f s t)
(hx : x ∈ s)
{C D : ℝ}
(hC : ∀ (i : ℕ), i ≤ n → ‖iterated_fderiv_within 𝕜 i g t (f x)‖ ≤ C)
(hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iterated_fderiv_within 𝕜 i f s x‖ ≤ D ^ i) :
If the derivatives within a set of g at f x are bounded by C, and the i-th derivative
within a set of f at x is bounded by D^i for all 1 ≤ i ≤ n, then the n-th derivative
of g ∘ f is bounded by n! * C * D^n.
This lemma proves this estimate assuming additionally that two of the spaces live in the same
universe, to make an induction possible. Use instead norm_iterated_fderiv_within_comp_le that
removes this assumption.
theorem
norm_iterated_fderiv_within_comp_le
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{g : F → G}
{f : E → F}
{n : ℕ}
{s : set E}
{t : set F}
{x : E}
{N : ℕ∞}
(hg : cont_diff_on 𝕜 N g t)
(hf : cont_diff_on 𝕜 N f s)
(hn : ↑n ≤ N)
(ht : unique_diff_on 𝕜 t)
(hs : unique_diff_on 𝕜 s)
(hst : set.maps_to f s t)
(hx : x ∈ s)
{C D : ℝ}
(hC : ∀ (i : ℕ), i ≤ n → ‖iterated_fderiv_within 𝕜 i g t (f x)‖ ≤ C)
(hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iterated_fderiv_within 𝕜 i f s x‖ ≤ D ^ i) :
If the derivatives within a set of g at f x are bounded by C, and the i-th derivative
within a set of f at x is bounded by D^i for all 1 ≤ i ≤ n, then the n-th derivative
of g ∘ f is bounded by n! * C * D^n.
theorem
norm_iterated_fderiv_comp_le
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{g : F → G}
{f : E → F}
{n : ℕ}
{N : ℕ∞}
(hg : cont_diff 𝕜 N g)
(hf : cont_diff 𝕜 N f)
(hn : ↑n ≤ N)
(x : E)
{C D : ℝ}
(hC : ∀ (i : ℕ), i ≤ n → ‖iterated_fderiv 𝕜 i g (f x)‖ ≤ C)
(hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iterated_fderiv 𝕜 i f x‖ ≤ D ^ i) :
If the derivatives of g at f x are bounded by C, and the i-th derivative
of f at x is bounded by D^i for all 1 ≤ i ≤ n, then the n-th derivative
of g ∘ f is bounded by n! * C * D^n.
theorem
norm_iterated_fderiv_within_clm_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{f : E → (F →L[𝕜] G)}
{g : E → F}
{s : set E}
{x : E}
{N : ℕ∞}
{n : ℕ}
(hf : cont_diff_on 𝕜 N f s)
(hg : cont_diff_on 𝕜 N g s)
(hs : unique_diff_on 𝕜 s)
(hx : x ∈ s)
(hn : ↑n ≤ N) :
‖iterated_fderiv_within 𝕜 n (λ (y : E), ⇑(f y) (g y)) s x‖ ≤ (finset.range (n + 1)).sum (λ (i : ℕ), ↑(n.choose i) * ‖iterated_fderiv_within 𝕜 i f s x‖ * ‖iterated_fderiv_within 𝕜 (n - i) g s x‖)
theorem
norm_iterated_fderiv_clm_apply
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{f : E → (F →L[𝕜] G)}
{g : E → F}
{N : ℕ∞}
{n : ℕ}
(hf : cont_diff 𝕜 N f)
(hg : cont_diff 𝕜 N g)
(x : E)
(hn : ↑n ≤ N) :
‖iterated_fderiv 𝕜 n (λ (y : E), ⇑(f y) (g y)) x‖ ≤ (finset.range (n + 1)).sum (λ (i : ℕ), ↑(n.choose i) * ‖iterated_fderiv 𝕜 i f x‖ * ‖iterated_fderiv 𝕜 (n - i) g x‖)
theorem
norm_iterated_fderiv_within_clm_apply_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{f : E → (F →L[𝕜] G)}
{c : F}
{s : set E}
{x : E}
{N : ℕ∞}
{n : ℕ}
(hf : cont_diff_on 𝕜 N f s)
(hs : unique_diff_on 𝕜 s)
(hx : x ∈ s)
(hn : ↑n ≤ N) :
theorem
norm_iterated_fderiv_clm_apply_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type uE}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{F : Type uF}
[normed_add_comm_group F]
[normed_space 𝕜 F]
{G : Type uG}
[normed_add_comm_group G]
[normed_space 𝕜 G]
{f : E → (F →L[𝕜] G)}
{c : F}
{x : E}
{N : ℕ∞}
{n : ℕ}
(hf : cont_diff 𝕜 N f)
(hn : ↑n ≤ N) :