Operator norm on the space of continuous linear maps #
Define the operator norm on the space of continuous (semi)linear maps between normed spaces, and prove its basic properties. In particular, show that this space is itself a normed space.
Since a lot of elementary properties don't require ‖x‖ = 0 → x = 0 we start setting up the
theory for SeminormedAddCommGroup and we specialize to NormedAddCommGroup at the end.
Note that most of statements that apply to semilinear maps only hold when the ring homomorphism
is isometric, as expressed by the typeclass [RingHomIsometric σ].
theorem
norm_image_of_norm_zero
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
{𝓕 : Type u_10}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[SemilinearMapClass 𝓕 σ₁₂ E F]
(f : 𝓕)
(hf : Continuous ↑f)
{x : E}
(hx : ‖x‖ = 0)
:
If ‖x‖ = 0 and f is continuous then ‖f x‖ = 0.
theorem
SemilinearMapClass.bound_of_shell_semi_normed
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
{𝓕 : Type u_10}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
[SemilinearMapClass 𝓕 σ₁₂ E F]
(f : 𝓕)
{ε : ℝ}
{C : ℝ}
(ε_pos : 0 < ε)
{c : 𝕜}
(hc : 1 < ‖c‖)
(hf : ∀ (x : E), ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖↑f x‖ ≤ C * ‖x‖)
{x : E}
(hx : ‖x‖ ≠ 0)
:
theorem
SemilinearMapClass.bound_of_continuous
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
{𝓕 : Type u_10}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
[SemilinearMapClass 𝓕 σ₁₂ E F]
(f : 𝓕)
(hf : Continuous ↑f)
:
A continuous linear map between seminormed spaces is bounded when the field is nontrivially
normed. The continuity ensures boundedness on a ball of some radius ε. The nontriviality of the
norm is then used to rescale any element into an element of norm in [ε/C, ε], whose image has a
controlled norm. The norm control for the original element follows by rescaling.
theorem
ContinuousLinearMap.bound
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
:
def
LinearIsometry.toSpanSingleton
(𝕜 : Type u_1)
(E : Type u_4)
[SeminormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
{v : E}
(hv : ‖v‖ = 1)
:
Given a unit-length element x of a normed space E over a field 𝕜, the natural linear
isometry map from 𝕜 to E by taking multiples of x.
Instances For
@[simp]
theorem
LinearIsometry.toSpanSingleton_apply
{𝕜 : Type u_1}
{E : Type u_4}
[SeminormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
{v : E}
(hv : ‖v‖ = 1)
(a : 𝕜)
:
↑(LinearIsometry.toSpanSingleton 𝕜 E hv) a = a • v
@[simp]
theorem
LinearIsometry.coe_toSpanSingleton
{𝕜 : Type u_1}
{E : Type u_4}
[SeminormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
{v : E}
(hv : ‖v‖ = 1)
:
(LinearIsometry.toSpanSingleton 𝕜 E hv).toLinearMap = LinearMap.toSpanSingleton 𝕜 E v
def
ContinuousLinearMap.opNorm
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
(f : E →SL[σ₁₂] F)
:
The operator norm of a continuous linear map is the inf of all its bounds.
Instances For
instance
ContinuousLinearMap.hasOpNorm
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
:
theorem
ContinuousLinearMap.norm_def
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
(f : E →SL[σ₁₂] F)
:
theorem
ContinuousLinearMap.bounds_nonempty
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
{f : E →SL[σ₁₂] F}
:
theorem
ContinuousLinearMap.bounds_bddBelow
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{f : E →SL[σ₁₂] F}
:
theorem
ContinuousLinearMap.isLeast_op_norm
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
:
theorem
ContinuousLinearMap.op_norm_le_bound
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
(f : E →SL[σ₁₂] F)
{M : ℝ}
(hMp : 0 ≤ M)
(hM : ∀ (x : E), ‖↑f x‖ ≤ M * ‖x‖)
:
If one controls the norm of every A x, then one controls the norm of A.
theorem
ContinuousLinearMap.op_norm_le_bound'
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
(f : E →SL[σ₁₂] F)
{M : ℝ}
(hMp : 0 ≤ M)
(hM : ∀ (x : E), ‖x‖ ≠ 0 → ‖↑f x‖ ≤ M * ‖x‖)
:
If one controls the norm of every A x, ‖x‖ ≠ 0, then one controls the norm of A.
theorem
ContinuousLinearMap.op_norm_le_of_lipschitz
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{f : E →SL[σ₁₂] F}
{K : NNReal}
(hf : LipschitzWith K ↑f)
:
theorem
ContinuousLinearMap.op_norm_eq_of_bounds
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{φ : E →SL[σ₁₂] F}
{M : ℝ}
(M_nonneg : 0 ≤ M)
(h_above : ∀ (x : E), ‖↑φ x‖ ≤ M * ‖x‖)
(h_below : ∀ (N : ℝ), N ≥ 0 → (∀ (x : E), ‖↑φ x‖ ≤ N * ‖x‖) → M ≤ N)
:
theorem
ContinuousLinearMap.op_norm_neg
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
(f : E →SL[σ₁₂] F)
:
theorem
ContinuousLinearMap.op_norm_nonneg
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
(f : E →SL[σ₁₂] F)
:
theorem
ContinuousLinearMap.op_norm_zero
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
:
The norm of the 0 operator is 0.
theorem
ContinuousLinearMap.norm_id_le
{𝕜 : Type u_1}
{E : Type u_4}
[SeminormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
:
‖ContinuousLinearMap.id 𝕜 E‖ ≤ 1
The norm of the identity is at most 1. It is in fact 1, except when the space is trivial
where it is 0. It means that one can not do better than an inequality in general.
theorem
ContinuousLinearMap.le_op_norm
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
(x : E)
:
The fundamental property of the operator norm: ‖f x‖ ≤ ‖f‖ * ‖x‖.
theorem
ContinuousLinearMap.dist_le_op_norm
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
(x : E)
(y : E)
:
theorem
ContinuousLinearMap.le_of_op_norm_le_of_le
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
{x : E}
{a : ℝ}
{b : ℝ}
(hf : ‖f‖ ≤ a)
(hx : ‖x‖ ≤ b)
:
theorem
ContinuousLinearMap.le_op_norm_of_le
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
{c : ℝ}
{x : E}
(h : ‖x‖ ≤ c)
:
theorem
ContinuousLinearMap.le_of_op_norm_le
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
{c : ℝ}
(h : ‖f‖ ≤ c)
(x : E)
:
theorem
ContinuousLinearMap.op_norm_le_iff
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
{f : E →SL[σ₁₂] F}
{M : ℝ}
(hMp : 0 ≤ M)
:
theorem
ContinuousLinearMap.ratio_le_op_norm
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
(x : E)
:
theorem
ContinuousLinearMap.unit_le_op_norm
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
(x : E)
:
The image of the unit ball under a continuous linear map is bounded.
theorem
ContinuousLinearMap.op_norm_le_of_shell
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
{f : E →SL[σ₁₂] F}
{ε : ℝ}
{C : ℝ}
(ε_pos : 0 < ε)
(hC : 0 ≤ C)
{c : 𝕜}
(hc : 1 < ‖c‖)
(hf : ∀ (x : E), ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖↑f x‖ ≤ C * ‖x‖)
:
theorem
ContinuousLinearMap.op_norm_le_of_ball
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
{f : E →SL[σ₁₂] F}
{ε : ℝ}
{C : ℝ}
(ε_pos : 0 < ε)
(hC : 0 ≤ C)
(hf : ∀ (x : E), x ∈ Metric.ball 0 ε → ‖↑f x‖ ≤ C * ‖x‖)
:
theorem
ContinuousLinearMap.op_norm_le_of_nhds_zero
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
{f : E →SL[σ₁₂] F}
{C : ℝ}
(hC : 0 ≤ C)
(hf : ∀ᶠ (x : E) in nhds 0, ‖↑f x‖ ≤ C * ‖x‖)
:
theorem
ContinuousLinearMap.op_norm_le_of_shell'
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
{f : E →SL[σ₁₂] F}
{ε : ℝ}
{C : ℝ}
(ε_pos : 0 < ε)
(hC : 0 ≤ C)
{c : 𝕜}
(hc : ‖c‖ < 1)
(hf : ∀ (x : E), ε * ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖↑f x‖ ≤ C * ‖x‖)
:
theorem
ContinuousLinearMap.op_norm_le_of_unit_norm
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NormedSpace ℝ E]
[NormedSpace ℝ F]
{f : E →L[ℝ] F}
{C : ℝ}
(hC : 0 ≤ C)
(hf : ∀ (x : E), ‖x‖ = 1 → ‖↑f x‖ ≤ C)
:
For a continuous real linear map f, if one controls the norm of every f x, ‖x‖ = 1, then
one controls the norm of f.
theorem
ContinuousLinearMap.op_norm_add_le
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
(g : E →SL[σ₁₂] F)
:
The operator norm satisfies the triangle inequality.
theorem
ContinuousLinearMap.norm_id_of_nontrivial_seminorm
{𝕜 : Type u_1}
{E : Type u_4}
[SeminormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
(h : ∃ x, ‖x‖ ≠ 0)
:
‖ContinuousLinearMap.id 𝕜 E‖ = 1
If there is an element with norm different from 0, then the norm of the identity equals 1.
(Since we are working with seminorms supposing that the space is non-trivial is not enough.)
theorem
ContinuousLinearMap.op_norm_smul_le
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
{𝕜' : Type u_11}
[NormedField 𝕜']
[NormedSpace 𝕜' F]
[SMulCommClass 𝕜₂ 𝕜' F]
(c : 𝕜')
(f : E →SL[σ₁₂] F)
:
def
ContinuousLinearMap.tmpSeminormedAddCommGroup
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
:
SeminormedAddCommGroup (E →SL[σ₁₂] F)
Continuous linear maps themselves form a seminormed space with respect to
the operator norm. This is only a temporary definition because we want to replace the topology
with ContinuousLinearMap.topologicalSpace to avoid diamond issues.
See Note [forgetful inheritance]
Instances For
def
ContinuousLinearMap.tmpPseudoMetricSpace
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
:
PseudoMetricSpace (E →SL[σ₁₂] F)
The PseudoMetricSpace structure on E →SL[σ₁₂] F coming from
ContinuousLinearMap.tmpSeminormedAddCommGroup.
See Note [forgetful inheritance]
Instances For
def
ContinuousLinearMap.tmpUniformSpace
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
:
UniformSpace (E →SL[σ₁₂] F)
The UniformSpace structure on E →SL[σ₁₂] F coming from
ContinuousLinearMap.tmpSeminormedAddCommGroup.
See Note [forgetful inheritance]
Instances For
def
ContinuousLinearMap.tmpTopologicalSpace
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
:
TopologicalSpace (E →SL[σ₁₂] F)
The TopologicalSpace structure on E →SL[σ₁₂] F coming from
ContinuousLinearMap.tmpSeminormedAddCommGroup.
See Note [forgetful inheritance]
Instances For
theorem
ContinuousLinearMap.tmpTopologicalAddGroup
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
:
TopologicalAddGroup (E →SL[σ₁₂] F)
theorem
ContinuousLinearMap.tmp_closedBall_div_subset
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
{a : ℝ}
{b : ℝ}
(ha : 0 < a)
(hb : 0 < b)
:
Metric.closedBall 0 (a / b) ⊆ {f | ∀ (x : E), x ∈ Metric.closedBall 0 b → ↑f x ∈ Metric.closedBall 0 a}
theorem
ContinuousLinearMap.tmp_topology_eq
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
:
ContinuousLinearMap.tmpTopologicalSpace = ContinuousLinearMap.topologicalSpace
theorem
ContinuousLinearMap.tmpUniformSpace_eq
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
:
ContinuousLinearMap.tmpUniformSpace = ContinuousLinearMap.uniformSpace
instance
ContinuousLinearMap.toPseudoMetricSpace
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
:
PseudoMetricSpace (E →SL[σ₁₂] F)
instance
ContinuousLinearMap.toSeminormedAddCommGroup
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
:
SeminormedAddCommGroup (E →SL[σ₁₂] F)
Continuous linear maps themselves form a seminormed space with respect to the operator norm.
theorem
ContinuousLinearMap.nnnorm_def
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
:
theorem
ContinuousLinearMap.op_nnnorm_le_bound
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
(M : NNReal)
(hM : ∀ (x : E), ‖↑f x‖₊ ≤ M * ‖x‖₊)
:
If one controls the norm of every A x, then one controls the norm of A.
theorem
ContinuousLinearMap.op_nnnorm_le_bound'
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
(M : NNReal)
(hM : ∀ (x : E), ‖x‖₊ ≠ 0 → ‖↑f x‖₊ ≤ M * ‖x‖₊)
:
If one controls the norm of every A x, ‖x‖₊ ≠ 0, then one controls the norm of A.
theorem
ContinuousLinearMap.op_nnnorm_le_of_unit_nnnorm
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NormedSpace ℝ E]
[NormedSpace ℝ F]
{f : E →L[ℝ] F}
{C : NNReal}
(hf : ∀ (x : E), ‖x‖₊ = 1 → ‖↑f x‖₊ ≤ C)
:
For a continuous real linear map f, if one controls the norm of every f x, ‖x‖₊ = 1, then
one controls the norm of f.
theorem
ContinuousLinearMap.op_nnnorm_le_of_lipschitz
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
{f : E →SL[σ₁₂] F}
{K : NNReal}
(hf : LipschitzWith K ↑f)
:
theorem
ContinuousLinearMap.op_nnnorm_eq_of_bounds
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
{φ : E →SL[σ₁₂] F}
(M : NNReal)
(h_above : ∀ (x : E), ‖↑φ x‖ ≤ ↑M * ‖x‖)
(h_below : ∀ (N : NNReal), (∀ (x : E), ‖↑φ x‖₊ ≤ N * ‖x‖₊) → M ≤ N)
:
theorem
ContinuousLinearMap.op_nnnorm_le_iff
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
{f : E →SL[σ₁₂] F}
{C : NNReal}
:
theorem
ContinuousLinearMap.isLeast_op_nnnorm
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
:
instance
ContinuousLinearMap.toNormedSpace
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
{𝕜' : Type u_11}
[NormedField 𝕜']
[NormedSpace 𝕜' F]
[SMulCommClass 𝕜₂ 𝕜' F]
:
NormedSpace 𝕜' (E →SL[σ₁₂] F)
theorem
ContinuousLinearMap.op_norm_comp_le
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
[RingHomIsometric σ₁₂]
[RingHomIsometric σ₂₃]
(h : F →SL[σ₂₃] G)
(f : E →SL[σ₁₂] F)
:
The operator norm is submultiplicative.
theorem
ContinuousLinearMap.op_norm_comp_le'
{𝕜 : Type u_1}
{E : Type u_4}
{Eₗ : Type u_5}
{Fₗ : Type u_7}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Eₗ]
[SeminormedAddCommGroup Fₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Eₗ]
[NormedSpace 𝕜 Fₗ]
(h : Eₗ →L[𝕜] Fₗ)
(f : E →L[𝕜] Eₗ)
:
The operator norm is submultiplicative.
theorem
ContinuousLinearMap.op_nnnorm_comp_le
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
[RingHomIsometric σ₁₂]
[RingHomIsometric σ₂₃]
(h : F →SL[σ₂₃] G)
[RingHomIsometric σ₁₃]
(f : E →SL[σ₁₂] F)
:
instance
ContinuousLinearMap.toSemiNormedRing
{𝕜 : Type u_1}
{E : Type u_4}
[SeminormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
:
SeminormedRing (E →L[𝕜] E)
Continuous linear maps form a seminormed ring with respect to the operator norm.
instance
ContinuousLinearMap.toNormedAlgebra
{𝕜 : Type u_1}
{E : Type u_4}
[SeminormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
:
NormedAlgebra 𝕜 (E →L[𝕜] E)
For a normed space E, continuous linear endomorphisms form a normed algebra with
respect to the operator norm.
theorem
ContinuousLinearMap.le_op_nnnorm
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
(x : E)
:
theorem
ContinuousLinearMap.nndist_le_op_nnnorm
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
(x : E)
(y : E)
:
theorem
ContinuousLinearMap.lipschitz
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
:
LipschitzWith ‖f‖₊ ↑f
continuous linear maps are Lipschitz continuous.
theorem
ContinuousLinearMap.lipschitz_apply
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(x : E)
:
LipschitzWith ‖x‖₊ fun f => ↑f x
Evaluation of a continuous linear map f at a point is Lipschitz continuous in f.
theorem
ContinuousLinearMap.exists_mul_lt_apply_of_lt_op_nnnorm
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
{r : NNReal}
(hr : r < ‖f‖₊)
:
theorem
ContinuousLinearMap.exists_mul_lt_of_lt_op_norm
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
{r : ℝ}
(hr₀ : 0 ≤ r)
(hr : r < ‖f‖)
:
theorem
ContinuousLinearMap.exists_lt_apply_of_lt_op_nnnorm
{𝕜 : Type u_11}
{𝕜₂ : Type u_12}
{E : Type u_13}
{F : Type u_14}
[NormedAddCommGroup E]
[SeminormedAddCommGroup F]
[DenselyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
{r : NNReal}
(hr : r < ‖f‖₊)
:
theorem
ContinuousLinearMap.exists_lt_apply_of_lt_op_norm
{𝕜 : Type u_11}
{𝕜₂ : Type u_12}
{E : Type u_13}
{F : Type u_14}
[NormedAddCommGroup E]
[SeminormedAddCommGroup F]
[DenselyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
{r : ℝ}
(hr : r < ‖f‖)
:
theorem
ContinuousLinearMap.sSup_unit_ball_eq_nnnorm
{𝕜 : Type u_11}
{𝕜₂ : Type u_12}
{E : Type u_13}
{F : Type u_14}
[NormedAddCommGroup E]
[SeminormedAddCommGroup F]
[DenselyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
:
theorem
ContinuousLinearMap.sSup_unit_ball_eq_norm
{𝕜 : Type u_11}
{𝕜₂ : Type u_12}
{E : Type u_13}
{F : Type u_14}
[NormedAddCommGroup E]
[SeminormedAddCommGroup F]
[DenselyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
:
theorem
ContinuousLinearMap.sSup_closed_unit_ball_eq_nnnorm
{𝕜 : Type u_11}
{𝕜₂ : Type u_12}
{E : Type u_13}
{F : Type u_14}
[NormedAddCommGroup E]
[SeminormedAddCommGroup F]
[DenselyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
:
theorem
ContinuousLinearMap.sSup_closed_unit_ball_eq_norm
{𝕜 : Type u_11}
{𝕜₂ : Type u_12}
{E : Type u_13}
{F : Type u_14}
[NormedAddCommGroup E]
[SeminormedAddCommGroup F]
[DenselyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
:
theorem
ContinuousLinearMap.op_norm_ext
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₁₃]
(f : E →SL[σ₁₂] F)
(g : E →SL[σ₁₃] G)
(h : ∀ (x : E), ‖↑f x‖ = ‖↑g x‖)
:
theorem
ContinuousLinearMap.op_norm_le_bound₂
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
(f : E →SL[σ₁₃] F →SL[σ₂₃] G)
{C : ℝ}
(h0 : 0 ≤ C)
(hC : ∀ (x : E) (y : F), ‖↑(↑f x) y‖ ≤ C * ‖x‖ * ‖y‖)
:
theorem
ContinuousLinearMap.le_op_norm₂
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
(f : E →SL[σ₁₃] F →SL[σ₂₃] G)
(x : E)
(y : F)
:
theorem
ContinuousLinearMap.le_of_op_norm₂_le_of_le
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
(f : E →SL[σ₁₃] F →SL[σ₂₃] G)
{x : E}
{y : F}
{a : ℝ}
{b : ℝ}
{c : ℝ}
(hf : ‖f‖ ≤ a)
(hx : ‖x‖ ≤ b)
(hy : ‖y‖ ≤ c)
:
@[simp]
theorem
ContinuousLinearMap.op_norm_prod
{𝕜 : Type u_1}
{E : Type u_4}
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
(f : E →L[𝕜] Fₗ)
(g : E →L[𝕜] Gₗ)
:
@[simp]
theorem
ContinuousLinearMap.op_nnnorm_prod
{𝕜 : Type u_1}
{E : Type u_4}
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
(f : E →L[𝕜] Fₗ)
(g : E →L[𝕜] Gₗ)
:
def
ContinuousLinearMap.prodₗᵢ
{𝕜 : Type u_1}
{E : Type u_4}
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
(R : Type u_11)
[Semiring R]
[Module R Fₗ]
[Module R Gₗ]
[ContinuousConstSMul R Fₗ]
[ContinuousConstSMul R Gₗ]
[SMulCommClass 𝕜 R Fₗ]
[SMulCommClass 𝕜 R Gₗ]
:
ContinuousLinearMap.prod as a LinearIsometryEquiv.
Instances For
@[simp]
theorem
ContinuousLinearMap.op_norm_subsingleton
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
[Subsingleton E]
:
theorem
ContinuousLinearMap.isBigOWith_id
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
(l : Filter E)
:
Asymptotics.IsBigOWith ‖f‖ l ↑f fun x => x
theorem
ContinuousLinearMap.isBigO_id
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
(l : Filter E)
:
theorem
ContinuousLinearMap.isBigOWith_comp
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
[RingHomIsometric σ₂₃]
{α : Type u_11}
(g : F →SL[σ₂₃] G)
(f : α → F)
(l : Filter α)
:
Asymptotics.IsBigOWith ‖g‖ l (fun x' => ↑g (f x')) f
theorem
ContinuousLinearMap.isBigO_comp
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
[RingHomIsometric σ₂₃]
{α : Type u_11}
(g : F →SL[σ₂₃] G)
(f : α → F)
(l : Filter α)
:
theorem
ContinuousLinearMap.isBigOWith_sub
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
(l : Filter E)
(x : E)
:
Asymptotics.IsBigOWith ‖f‖ l (fun x' => ↑f (x' - x)) fun x' => x' - x
theorem
ContinuousLinearMap.isBigO_sub
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F)
(l : Filter E)
(x : E)
:
theorem
LinearIsometry.norm_toContinuousLinearMap_le
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
(f : E →ₛₗᵢ[σ₁₂] F)
:
theorem
LinearMap.mkContinuous_norm_le
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
(f : E →ₛₗ[σ₁₂] F)
{C : ℝ}
(hC : 0 ≤ C)
(h : ∀ (x : E), ‖↑f x‖ ≤ C * ‖x‖)
:
‖LinearMap.mkContinuous f C h‖ ≤ C
If a continuous linear map is constructed from a linear map via the constructor mkContinuous,
then its norm is bounded by the bound given to the constructor if it is nonnegative.
theorem
LinearMap.mkContinuous_norm_le'
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
(f : E →ₛₗ[σ₁₂] F)
{C : ℝ}
(h : ∀ (x : E), ‖↑f x‖ ≤ C * ‖x‖)
:
‖LinearMap.mkContinuous f C h‖ ≤ max C 0
If a continuous linear map is constructed from a linear map via the constructor mkContinuous,
then its norm is bounded by the bound or zero if bound is negative.
theorem
LinearMap.norm_mkContinuous₂_aux
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
(f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G)
(C : ℝ)
(h : ∀ (x : E) (y : F), ‖↑(↑f x) y‖ ≤ C * ‖x‖ * ‖y‖)
(x : E)
:
def
LinearMap.mkContinuousOfExistsBound₂
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
(f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G)
(h : ∃ C, ∀ (x : E) (y : F), ‖↑(↑f x) y‖ ≤ C * ‖x‖ * ‖y‖)
:
Create a bilinear map (represented as a map E →L[𝕜] F →L[𝕜] G) from the corresponding linear
map and existence of a bound on the norm of the image. The linear map can be constructed using
LinearMap.mk₂.
If you have an explicit bound, use LinearMap.mkContinuous₂ instead, as a norm estimate will
follow automatically in LinearMap.mkContinuous₂_norm_le.
Instances For
def
LinearMap.mkContinuous₂
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
(f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G)
(C : ℝ)
(hC : ∀ (x : E) (y : F), ‖↑(↑f x) y‖ ≤ C * ‖x‖ * ‖y‖)
:
Create a bilinear map (represented as a map E →L[𝕜] F →L[𝕜] G) from the corresponding linear
map and a bound on the norm of the image. The linear map can be constructed using
LinearMap.mk₂. Lemmas LinearMap.mkContinuous₂_norm_le' and LinearMap.mkContinuous₂_norm_le
provide estimates on the norm of an operator constructed using this function.
Instances For
@[simp]
theorem
LinearMap.mkContinuous₂_apply
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
(f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G)
{C : ℝ}
(hC : ∀ (x : E) (y : F), ‖↑(↑f x) y‖ ≤ C * ‖x‖ * ‖y‖)
(x : E)
(y : F)
:
↑(↑(LinearMap.mkContinuous₂ f C hC) x) y = ↑(↑f x) y
theorem
LinearMap.mkContinuous₂_norm_le'
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
(f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G)
{C : ℝ}
(hC : ∀ (x : E) (y : F), ‖↑(↑f x) y‖ ≤ C * ‖x‖ * ‖y‖)
:
‖LinearMap.mkContinuous₂ f C hC‖ ≤ max C 0
theorem
LinearMap.mkContinuous₂_norm_le
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
(f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G)
{C : ℝ}
(h0 : 0 ≤ C)
(hC : ∀ (x : E) (y : F), ‖↑(↑f x) y‖ ≤ C * ‖x‖ * ‖y‖)
:
‖LinearMap.mkContinuous₂ f C hC‖ ≤ C
def
ContinuousLinearMap.flip
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
(f : E →SL[σ₁₃] F →SL[σ₂₃] G)
:
Flip the order of arguments of a continuous bilinear map.
For a version bundled as LinearIsometryEquiv, see
ContinuousLinearMap.flipL.
Instances For
@[simp]
theorem
ContinuousLinearMap.flip_apply
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
(f : E →SL[σ₁₃] F →SL[σ₂₃] G)
(x : E)
(y : F)
:
↑(↑(ContinuousLinearMap.flip f) y) x = ↑(↑f x) y
@[simp]
theorem
ContinuousLinearMap.flip_flip
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
(f : E →SL[σ₁₃] F →SL[σ₂₃] G)
:
@[simp]
theorem
ContinuousLinearMap.op_norm_flip
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
(f : E →SL[σ₁₃] F →SL[σ₂₃] G)
:
@[simp]
theorem
ContinuousLinearMap.flip_add
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
(f : E →SL[σ₁₃] F →SL[σ₂₃] G)
(g : E →SL[σ₁₃] F →SL[σ₂₃] G)
:
@[simp]
theorem
ContinuousLinearMap.flip_smul
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
(c : 𝕜₃)
(f : E →SL[σ₁₃] F →SL[σ₂₃] G)
:
ContinuousLinearMap.flip (c • f) = c • ContinuousLinearMap.flip f
def
ContinuousLinearMap.flipₗᵢ'
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
(E : Type u_4)
(F : Type u_6)
(G : Type u_8)
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
(σ₂₃ : 𝕜₂ →+* 𝕜₃)
(σ₁₃ : 𝕜 →+* 𝕜₃)
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
:
Flip the order of arguments of a continuous bilinear map.
This is a version bundled as a LinearIsometryEquiv.
For an unbundled version see ContinuousLinearMap.flip.
Instances For
@[simp]
theorem
ContinuousLinearMap.flipₗᵢ'_symm
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
:
LinearIsometryEquiv.symm (ContinuousLinearMap.flipₗᵢ' E F G σ₂₃ σ₁₃) = ContinuousLinearMap.flipₗᵢ' F E G σ₁₃ σ₂₃
@[simp]
theorem
ContinuousLinearMap.coe_flipₗᵢ'
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
:
↑(ContinuousLinearMap.flipₗᵢ' E F G σ₂₃ σ₁₃) = ContinuousLinearMap.flip
def
ContinuousLinearMap.flipₗᵢ
(𝕜 : Type u_1)
(E : Type u_4)
(Fₗ : Type u_7)
(Gₗ : Type u_9)
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
:
Flip the order of arguments of a continuous bilinear map.
This is a version bundled as a LinearIsometryEquiv.
For an unbundled version see ContinuousLinearMap.flip.
Instances For
@[simp]
theorem
ContinuousLinearMap.flipₗᵢ_symm
{𝕜 : Type u_1}
{E : Type u_4}
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
:
LinearIsometryEquiv.symm (ContinuousLinearMap.flipₗᵢ 𝕜 E Fₗ Gₗ) = ContinuousLinearMap.flipₗᵢ 𝕜 Fₗ E Gₗ
@[simp]
theorem
ContinuousLinearMap.coe_flipₗᵢ
{𝕜 : Type u_1}
{E : Type u_4}
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
:
↑(ContinuousLinearMap.flipₗᵢ 𝕜 E Fₗ Gₗ) = ContinuousLinearMap.flip
noncomputable def
ContinuousLinearMap.apply'
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
(F : Type u_6)
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
(σ₁₂ : 𝕜 →+* 𝕜₂)
[RingHomIsometric σ₁₂]
:
The continuous semilinear map obtained by applying a continuous semilinear map at a given vector.
This is the continuous version of LinearMap.applyₗ.
Instances For
@[simp]
theorem
ContinuousLinearMap.apply_apply'
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(v : E)
(f : E →SL[σ₁₂] F)
:
↑(↑(ContinuousLinearMap.apply' F σ₁₂) v) f = ↑f v
noncomputable def
ContinuousLinearMap.apply
(𝕜 : Type u_1)
{E : Type u_4}
(Fₗ : Type u_7)
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
:
The continuous semilinear map obtained by applying a continuous semilinear map at a given vector.
This is the continuous version of LinearMap.applyₗ.
Instances For
@[simp]
theorem
ContinuousLinearMap.apply_apply
{𝕜 : Type u_1}
{E : Type u_4}
{Fₗ : Type u_7}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
(v : E)
(f : E →L[𝕜] Fₗ)
:
↑(↑(ContinuousLinearMap.apply 𝕜 Fₗ) v) f = ↑f v
noncomputable def
ContinuousLinearMap.compSL
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
(E : Type u_4)
(F : Type u_6)
(G : Type u_8)
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
(σ₁₂ : 𝕜 →+* 𝕜₂)
(σ₂₃ : 𝕜₂ →+* 𝕜₃)
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
[RingHomIsometric σ₁₂]
:
Composition of continuous semilinear maps as a continuous semibilinear map.
Instances For
def
ContinuousLinearMap.instNormContinuousLinearMapToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldIdToNonAssocSemiringContinuousLinearMapToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToModuleToModuleTopologicalSpaceToTopologicalAddGroupAddCommMonoidToContinuousAddToAddGroupToSeminormedAddGroupContinuousLinearMapContinuousLinearMapToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToModuleTopologicalSpaceToTopologicalAddGroupAddCommMonoidToContinuousAddToAddGroupToSeminormedAddGroupContinuousLinearMapTopologicalSpaceAddCommMonoidModuleSmulCommClass_selfToCommMonoidToCommRingToEuclideanDomainToMulActionToMonoidWithZeroToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToMulActionWithZeroContinuousConstSMulToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingToSMulToZeroToAddMonoidToSMulZeroClassToZeroToSMulWithZeroContinuousSMulToZeroToCommMonoidWithZeroToCommGroupWithZeroBoundedSMulModuleSmulCommClass_selfToCommMonoidToCommRingToEuclideanDomainToMulActionToMonoidWithZeroToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToMulActionWithZeroContinuousConstSMulToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingToSMulToZeroToAddMonoidToSMulZeroClassToZeroToSMulWithZeroContinuousSMulToZeroToCommMonoidWithZeroToCommGroupWithZeroBoundedSMulTopologicalSpaceAddCommGroupToRingToNormedRingToNormedCommRingToRingToNormedRingAddCommGroupToRingToNormedRingTopologicalAddGroupAddCommMonoidToContinuousAddToAddGroupToSeminormedAddGroupToSeminormedAddCommGroupModuleModuleSmulCommClassToMonoidToDistribMulActionToSMulZeroClassToAddZeroClassToDistribSMulContinuousConstSMulToSMulToZeroToAddMonoidToSMulZeroClassToSMulWithZeroToMulActionWithZeroContinuousSMulToPseudoMetricSpaceZeroBoundedSMulToNormedSpace
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
(E : Type u_4)
(F : Type u_6)
(G : Type u_8)
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
(σ₁₂ : 𝕜 →+* 𝕜₂)
(σ₂₃ : 𝕜₂ →+* 𝕜₃)
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
[RingHomIsometric σ₁₂]
:
Porting note: Local instance for norm_compSL_le.
Should be by inferInstance, and indeed not be needed.
Instances For
theorem
ContinuousLinearMap.norm_compSL_le
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
(E : Type u_4)
(F : Type u_6)
(G : Type u_8)
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
(σ₁₂ : 𝕜 →+* 𝕜₂)
(σ₂₃ : 𝕜₂ →+* 𝕜₃)
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
[RingHomIsometric σ₁₂]
:
‖ContinuousLinearMap.compSL E F G σ₁₂ σ₂₃‖ ≤ 1
@[simp]
theorem
ContinuousLinearMap.compSL_apply
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
[RingHomIsometric σ₁₂]
(f : F →SL[σ₂₃] G)
(g : E →SL[σ₁₂] F)
:
↑(↑(ContinuousLinearMap.compSL E F G σ₁₂ σ₂₃) f) g = ContinuousLinearMap.comp f g
theorem
Continuous.const_clm_comp
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
[RingHomIsometric σ₁₂]
{X : Type u_11}
[TopologicalSpace X]
{f : X → E →SL[σ₁₂] F}
(hf : Continuous f)
(g : F →SL[σ₂₃] G)
:
Continuous fun x => ContinuousLinearMap.comp g (f x)
theorem
Continuous.clm_comp_const
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
[RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃]
[RingHomIsometric σ₁₂]
{X : Type u_11}
[TopologicalSpace X]
{g : X → F →SL[σ₂₃] G}
(hg : Continuous g)
(f : E →SL[σ₁₂] F)
:
Continuous fun x => ContinuousLinearMap.comp (g x) f
def
ContinuousLinearMap.compL
(𝕜 : Type u_1)
(E : Type u_4)
(Fₗ : Type u_7)
(Gₗ : Type u_9)
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
:
Composition of continuous linear maps as a continuous bilinear map.
Instances For
def
ContinuousLinearMap.instNormContinuousLinearMapToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldIdToNonAssocSemiringContinuousLinearMapToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToModuleToModuleTopologicalSpaceToTopologicalAddGroupAddCommMonoidToContinuousAddToAddGroupToSeminormedAddGroupContinuousLinearMapContinuousLinearMapToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToModuleTopologicalSpaceToTopologicalAddGroupAddCommMonoidToContinuousAddToAddGroupToSeminormedAddGroupContinuousLinearMapTopologicalSpaceAddCommMonoidModuleSmulCommClass_selfToCommMonoidToCommRingToEuclideanDomainToMulActionToMonoidWithZeroToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToMulActionWithZeroContinuousConstSMulToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingToSMulToZeroToAddMonoidToSMulZeroClassToZeroToSMulWithZeroContinuousSMulToZeroToCommMonoidWithZeroToCommGroupWithZeroBoundedSMulModuleSmulCommClass_selfToMulActionToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToMulActionWithZeroContinuousConstSMulToSMulToZeroToAddMonoidToSMulZeroClassToSMulWithZeroContinuousSMulBoundedSMulTopologicalSpaceAddCommGroupToRingToNormedRingAddCommGroupTopologicalAddGroupAddCommMonoidToContinuousAddToAddGroupToSeminormedAddGroupToSeminormedAddCommGroupIdsModuleModuleSmulCommClassToMonoidToDistribMulActionToSMulZeroClassToAddZeroClassToDistribSMulContinuousConstSMulToSMulToZeroToAddMonoidToSMulZeroClassToSMulWithZeroToMulActionWithZeroContinuousSMulIds
(𝕜 : Type u_1)
(E : Type u_4)
(Fₗ : Type u_7)
(Gₗ : Type u_9)
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
:
Porting note: Local instance for norm_compL_le.
Should be by inferInstance, and indeed not be needed.
Instances For
theorem
ContinuousLinearMap.norm_compL_le
(𝕜 : Type u_1)
(E : Type u_4)
(Fₗ : Type u_7)
(Gₗ : Type u_9)
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
:
‖ContinuousLinearMap.compL 𝕜 E Fₗ Gₗ‖ ≤ 1
@[simp]
theorem
ContinuousLinearMap.compL_apply
(𝕜 : Type u_1)
(E : Type u_4)
(Fₗ : Type u_7)
(Gₗ : Type u_9)
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
(f : Fₗ →L[𝕜] Gₗ)
(g : E →L[𝕜] Fₗ)
:
↑(↑(ContinuousLinearMap.compL 𝕜 E Fₗ Gₗ) f) g = ContinuousLinearMap.comp f g
@[simp]
theorem
ContinuousLinearMap.precompR_apply
{𝕜 : Type u_1}
{E : Type u_4}
(Eₗ : Type u_5)
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Eₗ]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Eₗ]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
(L : E →L[𝕜] Fₗ →L[𝕜] Gₗ)
:
∀ (a : E), ↑(ContinuousLinearMap.precompR Eₗ L) a = ↑(ContinuousLinearMap.compL 𝕜 Eₗ Fₗ Gₗ) (↑L a)
def
ContinuousLinearMap.precompR
{𝕜 : Type u_1}
{E : Type u_4}
(Eₗ : Type u_5)
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Eₗ]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Eₗ]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
(L : E →L[𝕜] Fₗ →L[𝕜] Gₗ)
:
Apply L(x,-) pointwise to bilinear maps, as a continuous bilinear map
Instances For
def
ContinuousLinearMap.precompL
{𝕜 : Type u_1}
{E : Type u_4}
(Eₗ : Type u_5)
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Eₗ]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Eₗ]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
(L : E →L[𝕜] Fₗ →L[𝕜] Gₗ)
:
Apply L(-,y) pointwise to bilinear maps, as a continuous bilinear map
Instances For
def
ContinuousLinearMap.instSeminormedAddCommGroupContinuousLinearMapToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldIdToNonAssocSemiringContinuousLinearMapToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToModuleToModuleTopologicalSpaceToTopologicalAddGroupAddCommMonoidToContinuousAddToAddGroupToSeminormedAddGroupContinuousLinearMapToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToModuleTopologicalSpaceToTopologicalAddGroupAddCommMonoidToContinuousAddToAddGroupToSeminormedAddGroupModuleSmulCommClass_selfToCommMonoidToCommRingToEuclideanDomainToMulActionToMonoidWithZeroToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToMulActionWithZeroContinuousConstSMulToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingToSMulToZeroToAddMonoidToSMulZeroClassToZeroToSMulWithZeroContinuousSMulToZeroToCommMonoidWithZeroToCommGroupWithZeroBoundedSMulModuleSmulCommClass_selfToMulActionToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToMulActionWithZeroContinuousConstSMulToSMulToZeroToAddMonoidToSMulZeroClassToSMulWithZeroContinuousSMulBoundedSMul
{𝕜 : Type u_1}
(Eₗ : Type u_5)
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup Eₗ]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 Eₗ]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
:
Porting note: Local instances for norm_precompR_le.
Should be by inferInstance, and indeed not be needed.
Instances For
def
ContinuousLinearMap.instNormedSpaceContinuousLinearMapToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldIdToNonAssocSemiringContinuousLinearMapToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToModuleToModuleTopologicalSpaceToTopologicalAddGroupAddCommMonoidToContinuousAddToAddGroupToSeminormedAddGroupContinuousLinearMapToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToModuleTopologicalSpaceToTopologicalAddGroupAddCommMonoidToContinuousAddToAddGroupToSeminormedAddGroupModuleSmulCommClass_selfToCommMonoidToCommRingToEuclideanDomainToMulActionToMonoidWithZeroToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToMulActionWithZeroContinuousConstSMulToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingToSMulToZeroToAddMonoidToSMulZeroClassToZeroToSMulWithZeroContinuousSMulToZeroToCommMonoidWithZeroToCommGroupWithZeroBoundedSMulModuleSmulCommClass_selfToMulActionToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToMulActionWithZeroContinuousConstSMulToSMulToZeroToAddMonoidToSMulZeroClassToSMulWithZeroContinuousSMulBoundedSMulInstSeminormedAddCommGroupContinuousLinearMapToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldIdToNonAssocSemiringContinuousLinearMapToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToModuleToModuleTopologicalSpaceToTopologicalAddGroupAddCommMonoidToContinuousAddToAddGroupToSeminormedAddGroupContinuousLinearMapToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToModuleTopologicalSpaceToTopologicalAddGroupAddCommMonoidToContinuousAddToAddGroupToSeminormedAddGroupModuleSmulCommClass_selfToCommMonoidToCommRingToEuclideanDomainToMulActionToMonoidWithZeroToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToMulActionWithZeroContinuousConstSMulToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingToSMulToZeroToAddMonoidToSMulZeroClassToZeroToSMulWithZeroContinuousSMulToZeroToCommMonoidWithZeroToCommGroupWithZeroBoundedSMulModuleSmulCommClass_selfToMulActionToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToMulActionWithZeroContinuousConstSMulToSMulToZeroToAddMonoidToSMulZeroClassToSMulWithZeroContinuousSMulBoundedSMul
{𝕜 : Type u_1}
(Eₗ : Type u_5)
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup Eₗ]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 Eₗ]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
:
Instances For
theorem
ContinuousLinearMap.norm_precompR_le
{𝕜 : Type u_1}
{E : Type u_4}
(Eₗ : Type u_5)
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Eₗ]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Eₗ]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
(L : E →L[𝕜] Fₗ →L[𝕜] Gₗ)
:
def
ContinuousLinearMap.instNormContinuousLinearMapToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldIdToNonAssocSemiringContinuousLinearMapToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToModuleToModuleTopologicalSpaceToTopologicalAddGroupAddCommMonoidToContinuousAddToAddGroupToSeminormedAddGroupContinuousLinearMapToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupContinuousLinearMapToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToModuleTopologicalSpaceToTopologicalAddGroupAddCommMonoidToContinuousAddToAddGroupToSeminormedAddGroupToModuleModuleSmulCommClass_selfToCommMonoidToCommRingToEuclideanDomainToMulActionToMonoidWithZeroToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToMulActionWithZeroContinuousConstSMulToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingToSMulToZeroToAddMonoidToSMulZeroClassToZeroToSMulWithZeroContinuousSMulToZeroToCommMonoidWithZeroToCommGroupWithZeroBoundedSMulTopologicalSpaceAddCommGroupToRingToNormedRingTopologicalAddGroupAddCommMonoidToContinuousAddToAddGroupToSeminormedAddGroupToSeminormedAddCommGroupIdsModuleSmulCommClass_selfToMulActionToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToMulActionWithZeroContinuousConstSMulToSMulToZeroToAddMonoidToSMulZeroClassToSMulWithZeroContinuousSMulBoundedSMulModuleSmulCommClassToMonoidToDistribMulActionToSMulZeroClassToAddZeroClassToDistribSMulContinuousConstSMulToSMulToZeroToAddMonoidToSMulZeroClassToSMulWithZeroToMulActionWithZeroContinuousSMulIds
{𝕜 : Type u_1}
{E : Type u_4}
(Eₗ : Type u_5)
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Eₗ]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Eₗ]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
:
Porting note: Local instance for norm_precompL_le.
Should be by inferInstance, and indeed not be needed.
Instances For
theorem
ContinuousLinearMap.norm_precompL_le
{𝕜 : Type u_1}
{E : Type u_4}
(Eₗ : Type u_5)
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Eₗ]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Eₗ]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
(L : E →L[𝕜] Fₗ →L[𝕜] Gₗ)
:
def
ContinuousLinearMap.prodMapL
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(M₁ : Type u₁)
[SeminormedAddCommGroup M₁]
[NormedSpace 𝕜 M₁]
(M₂ : Type u₂)
[SeminormedAddCommGroup M₂]
[NormedSpace 𝕜 M₂]
(M₃ : Type u₃)
[SeminormedAddCommGroup M₃]
[NormedSpace 𝕜 M₃]
(M₄ : Type u₄)
[SeminormedAddCommGroup M₄]
[NormedSpace 𝕜 M₄]
:
ContinuousLinearMap.prodMap as a continuous linear map.
Instances For
@[simp]
theorem
ContinuousLinearMap.prodMapL_apply
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
{M₁ : Type u₁}
[SeminormedAddCommGroup M₁]
[NormedSpace 𝕜 M₁]
{M₂ : Type u₂}
[SeminormedAddCommGroup M₂]
[NormedSpace 𝕜 M₂]
{M₃ : Type u₃}
[SeminormedAddCommGroup M₃]
[NormedSpace 𝕜 M₃]
{M₄ : Type u₄}
[SeminormedAddCommGroup M₄]
[NormedSpace 𝕜 M₄]
(p : (M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄))
:
↑(ContinuousLinearMap.prodMapL 𝕜 M₁ M₂ M₃ M₄) p = ContinuousLinearMap.prodMap p.fst p.snd
theorem
Continuous.prod_mapL
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
{M₁ : Type u₁}
[SeminormedAddCommGroup M₁]
[NormedSpace 𝕜 M₁]
{M₂ : Type u₂}
[SeminormedAddCommGroup M₂]
[NormedSpace 𝕜 M₂]
{M₃ : Type u₃}
[SeminormedAddCommGroup M₃]
[NormedSpace 𝕜 M₃]
{M₄ : Type u₄}
[SeminormedAddCommGroup M₄]
[NormedSpace 𝕜 M₄]
{X : Type u_11}
[TopologicalSpace X]
{f : X → M₁ →L[𝕜] M₂}
{g : X → M₃ →L[𝕜] M₄}
(hf : Continuous f)
(hg : Continuous g)
:
Continuous fun x => ContinuousLinearMap.prodMap (f x) (g x)
theorem
Continuous.prod_map_equivL
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
{M₁ : Type u₁}
[SeminormedAddCommGroup M₁]
[NormedSpace 𝕜 M₁]
{M₂ : Type u₂}
[SeminormedAddCommGroup M₂]
[NormedSpace 𝕜 M₂]
{M₃ : Type u₃}
[SeminormedAddCommGroup M₃]
[NormedSpace 𝕜 M₃]
{M₄ : Type u₄}
[SeminormedAddCommGroup M₄]
[NormedSpace 𝕜 M₄]
{X : Type u_11}
[TopologicalSpace X]
{f : X → M₁ ≃L[𝕜] M₂}
{g : X → M₃ ≃L[𝕜] M₄}
(hf : Continuous fun x => ↑(f x))
(hg : Continuous fun x => ↑(g x))
:
Continuous fun x => ↑(ContinuousLinearEquiv.prod (f x) (g x))
theorem
ContinuousOn.prod_mapL
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
{M₁ : Type u₁}
[SeminormedAddCommGroup M₁]
[NormedSpace 𝕜 M₁]
{M₂ : Type u₂}
[SeminormedAddCommGroup M₂]
[NormedSpace 𝕜 M₂]
{M₃ : Type u₃}
[SeminormedAddCommGroup M₃]
[NormedSpace 𝕜 M₃]
{M₄ : Type u₄}
[SeminormedAddCommGroup M₄]
[NormedSpace 𝕜 M₄]
{X : Type u_11}
[TopologicalSpace X]
{f : X → M₁ →L[𝕜] M₂}
{g : X → M₃ →L[𝕜] M₄}
{s : Set X}
(hf : ContinuousOn f s)
(hg : ContinuousOn g s)
:
ContinuousOn (fun x => ContinuousLinearMap.prodMap (f x) (g x)) s
theorem
ContinuousOn.prod_map_equivL
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
{M₁ : Type u₁}
[SeminormedAddCommGroup M₁]
[NormedSpace 𝕜 M₁]
{M₂ : Type u₂}
[SeminormedAddCommGroup M₂]
[NormedSpace 𝕜 M₂]
{M₃ : Type u₃}
[SeminormedAddCommGroup M₃]
[NormedSpace 𝕜 M₃]
{M₄ : Type u₄}
[SeminormedAddCommGroup M₄]
[NormedSpace 𝕜 M₄]
{X : Type u_11}
[TopologicalSpace X]
{f : X → M₁ ≃L[𝕜] M₂}
{g : X → M₃ ≃L[𝕜] M₄}
{s : Set X}
(hf : ContinuousOn (fun x => ↑(f x)) s)
(hg : ContinuousOn (fun x => ↑(g x)) s)
:
ContinuousOn (fun x => ↑(ContinuousLinearEquiv.prod (f x) (g x))) s
def
ContinuousLinearMap.mul
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
:
Multiplication in a non-unital normed algebra as a continuous bilinear map.
Instances For
@[simp]
theorem
ContinuousLinearMap.mul_apply'
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
(x : 𝕜')
(y : 𝕜')
:
↑(↑(ContinuousLinearMap.mul 𝕜 𝕜') x) y = x * y
@[simp]
theorem
ContinuousLinearMap.op_norm_mul_apply_le
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
(x : 𝕜')
:
theorem
ContinuousLinearMap.op_norm_mul_le
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
:
‖ContinuousLinearMap.mul 𝕜 𝕜'‖ ≤ 1
def
NonUnitalAlgHom.Lmul
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
:
Multiplication on the left in a non-unital normed algebra 𝕜' as a non-unital algebra
homomorphism into the algebra of continuous linear maps. This is the left regular representation
of A acting on itself.
This has more algebraic structure than ContinuousLinearMap.mul, but there is no longer continuity
bundled in the first coordinate. An alternative viewpoint is that this upgrades
NonUnitalAlgHom.lmul from a homomorphism into linear maps to a homomorphism into continuous
linear maps.
Instances For
@[simp]
theorem
NonUnitalAlgHom.coe_Lmul
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{𝕜' : Type u_11}
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
:
↑(NonUnitalAlgHom.Lmul 𝕜 𝕜') = ↑(ContinuousLinearMap.mul 𝕜 𝕜')
def
ContinuousLinearMap.mulLeftRight
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
:
Simultaneous left- and right-multiplication in a non-unital normed algebra, considered as a
continuous trilinear map. This is akin to its non-continuous version LinearMap.mulLeftRight,
but there is a minor difference: LinearMap.mulLeftRight is uncurried.
Instances For
@[simp]
theorem
ContinuousLinearMap.mulLeftRight_apply
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
(x : 𝕜')
(y : 𝕜')
(z : 𝕜')
:
↑(↑(↑(ContinuousLinearMap.mulLeftRight 𝕜 𝕜') x) y) z = x * z * y
theorem
ContinuousLinearMap.op_norm_mulLeftRight_apply_apply_le
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
(x : 𝕜')
(y : 𝕜')
:
theorem
ContinuousLinearMap.op_norm_mulLeftRight_apply_le
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
(x : 𝕜')
:
def
ContinuousLinearMap.instNormContinuousLinearMapToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldIdToNonAssocSemiringToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToModuleToSeminormedAddCommGroupTopologicalSpaceToAddCommGroupToTopologicalAddGroupAddCommMonoidToContinuousAddToTopologicalSemiringSemi_normed_top_ringModuleSmulCommClass_selfToCommMonoidToCommRingToEuclideanDomainToMulActionToMonoidWithZeroToZeroToSemigroupWithZeroToNonUnitalSemiringToMulActionWithZeroContinuousConstSMulToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingToSMulToZeroToAddMonoidToSMulZeroClassToZeroToSMulWithZeroContinuousSMulToZeroToCommMonoidWithZeroToCommGroupWithZeroBoundedSMulAddCommGroupToRingToNormedRingTopologicalAddGroupToNonUnitalNonAssocRingToNonAssocRingRingToNonUnitalSeminormedRingToSemiNormedRingSmulCommClassToMonoidToDistribMulActionToSMulZeroClassToAddZeroClassToDistribSMulContinuousConstSMulToMonoidMonoidWithZeroSmulCommClassToDistribMulActionSemiringApplyModuleApplySMulCommClassContinuousConstSMulToContinuousMul
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
:
Porting note: Local instance for op_norm_mulLeftRight_le.
Should be by inferInstance, and indeed not be needed.
Instances For
theorem
ContinuousLinearMap.op_norm_mulLeftRight_le
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
:
class
RegularNormedAlgebra
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
:
- isometry_mul' : Isometry ↑(ContinuousLinearMap.mul 𝕜 𝕜')
The left regular representation of the algebra on itself is an isometry.
This is a mixin class for non-unital normed algebras which states that the left-regular
representation of the algebra on itself is isometric. Every unital normed algebra with ‖1‖ = 1 is
a regular normed algebra (see NormedAlgebra.instRegularNormedAlgebra). In addition, so is every
C⋆-algebra, non-unital included (see CstarRing.instRegularNormedAlgebra), but there are yet other
examples. Any algebra with an approximate identity (e.g., $$L^1$$) is also regular.
This is a useful class because it gives rise to a nice norm on the unitization; in particular it is
a C⋆-norm when the norm on A is a C⋆-norm.
Instances
instance
NormedAlgebra.instRegularNormedAlgebra
{𝕜 : Type u_12}
{𝕜' : Type u_13}
[NontriviallyNormedField 𝕜]
[SeminormedRing 𝕜']
[NormedAlgebra 𝕜 𝕜']
[NormOneClass 𝕜']
:
RegularNormedAlgebra 𝕜 𝕜'
Every (unital) normed algebra such that ‖1‖ = 1 is a RegularNormedAlgebra.
theorem
ContinuousLinearMap.isometry_mul
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
[RegularNormedAlgebra 𝕜 𝕜']
:
Isometry ↑(ContinuousLinearMap.mul 𝕜 𝕜')
@[simp]
theorem
ContinuousLinearMap.op_norm_mul_apply
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
[RegularNormedAlgebra 𝕜 𝕜']
(x : 𝕜')
:
@[simp]
theorem
ContinuousLinearMap.op_nnnorm_mul_apply
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
[RegularNormedAlgebra 𝕜 𝕜']
(x : 𝕜')
:
def
ContinuousLinearMap.mulₗᵢ
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
[RegularNormedAlgebra 𝕜 𝕜']
:
Multiplication in a normed algebra as a linear isometry to the space of continuous linear maps.
Instances For
@[simp]
theorem
ContinuousLinearMap.coe_mulₗᵢ
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_11)
[NonUnitalSeminormedRing 𝕜']
[NormedSpace 𝕜 𝕜']
[IsScalarTower 𝕜 𝕜' 𝕜']
[SMulCommClass 𝕜 𝕜' 𝕜']
[RegularNormedAlgebra 𝕜 𝕜']
:
↑(ContinuousLinearMap.mulₗᵢ 𝕜 𝕜') = ↑(ContinuousLinearMap.mul 𝕜 𝕜')
def
ContinuousLinearMap.lsmul
(𝕜 : Type u_1)
{E : Type u_4}
[SeminormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
(𝕜' : Type u_11)
[NormedField 𝕜']
[NormedAlgebra 𝕜 𝕜']
[NormedSpace 𝕜' E]
[IsScalarTower 𝕜 𝕜' E]
:
Scalar multiplication as a continuous bilinear map.
Instances For
@[simp]
theorem
ContinuousLinearMap.lsmul_apply
(𝕜 : Type u_1)
{E : Type u_4}
[SeminormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
(𝕜' : Type u_11)
[NormedField 𝕜']
[NormedAlgebra 𝕜 𝕜']
[NormedSpace 𝕜' E]
[IsScalarTower 𝕜 𝕜' E]
(c : 𝕜')
(x : E)
:
↑(↑(ContinuousLinearMap.lsmul 𝕜 𝕜') c) x = c • x
theorem
ContinuousLinearMap.norm_toSpanSingleton
(𝕜 : Type u_1)
{E : Type u_4}
[SeminormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
(x : E)
:
theorem
ContinuousLinearMap.op_norm_lsmul_apply_le
{𝕜 : Type u_1}
{E : Type u_4}
[SeminormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
{𝕜' : Type u_11}
[NormedField 𝕜']
[NormedAlgebra 𝕜 𝕜']
[NormedSpace 𝕜' E]
[IsScalarTower 𝕜 𝕜' E]
(x : 𝕜')
:
theorem
ContinuousLinearMap.op_norm_lsmul_le
{𝕜 : Type u_1}
{E : Type u_4}
[SeminormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
{𝕜' : Type u_11}
[NormedField 𝕜']
[NormedAlgebra 𝕜 𝕜']
[NormedSpace 𝕜' E]
[IsScalarTower 𝕜 𝕜' E]
:
‖ContinuousLinearMap.lsmul 𝕜 𝕜'‖ ≤ 1
The norm of lsmul is at most 1 in any semi-normed group.
@[simp]
theorem
ContinuousLinearMap.norm_restrictScalars
{𝕜 : Type u_1}
{E : Type u_4}
{Fₗ : Type u_7}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
{𝕜' : Type u_11}
[NontriviallyNormedField 𝕜']
[NormedAlgebra 𝕜' 𝕜]
[NormedSpace 𝕜' E]
[IsScalarTower 𝕜' 𝕜 E]
[NormedSpace 𝕜' Fₗ]
[IsScalarTower 𝕜' 𝕜 Fₗ]
(f : E →L[𝕜] Fₗ)
:
def
ContinuousLinearMap.restrictScalarsIsometry
(𝕜 : Type u_1)
(E : Type u_4)
(Fₗ : Type u_7)
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
(𝕜' : Type u_11)
[NontriviallyNormedField 𝕜']
[NormedAlgebra 𝕜' 𝕜]
[NormedSpace 𝕜' E]
[IsScalarTower 𝕜' 𝕜 E]
[NormedSpace 𝕜' Fₗ]
[IsScalarTower 𝕜' 𝕜 Fₗ]
(𝕜'' : Type u_12)
[Ring 𝕜'']
[Module 𝕜'' Fₗ]
[ContinuousConstSMul 𝕜'' Fₗ]
[SMulCommClass 𝕜 𝕜'' Fₗ]
[SMulCommClass 𝕜' 𝕜'' Fₗ]
:
ContinuousLinearMap.restrictScalars as a LinearIsometry.
Instances For
@[simp]
theorem
ContinuousLinearMap.coe_restrictScalarsIsometry
(𝕜 : Type u_1)
(E : Type u_4)
(Fₗ : Type u_7)
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
(𝕜' : Type u_11)
[NontriviallyNormedField 𝕜']
[NormedAlgebra 𝕜' 𝕜]
[NormedSpace 𝕜' E]
[IsScalarTower 𝕜' 𝕜 E]
[NormedSpace 𝕜' Fₗ]
[IsScalarTower 𝕜' 𝕜 Fₗ]
{𝕜'' : Type u_12}
[Ring 𝕜'']
[Module 𝕜'' Fₗ]
[ContinuousConstSMul 𝕜'' Fₗ]
[SMulCommClass 𝕜 𝕜'' Fₗ]
[SMulCommClass 𝕜' 𝕜'' Fₗ]
:
↑(ContinuousLinearMap.restrictScalarsIsometry 𝕜 E Fₗ 𝕜' 𝕜'') = ContinuousLinearMap.restrictScalars 𝕜'
@[simp]
theorem
ContinuousLinearMap.restrictScalarsIsometry_toLinearMap
(𝕜 : Type u_1)
(E : Type u_4)
(Fₗ : Type u_7)
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
(𝕜' : Type u_11)
[NontriviallyNormedField 𝕜']
[NormedAlgebra 𝕜' 𝕜]
[NormedSpace 𝕜' E]
[IsScalarTower 𝕜' 𝕜 E]
[NormedSpace 𝕜' Fₗ]
[IsScalarTower 𝕜' 𝕜 Fₗ]
{𝕜'' : Type u_12}
[Ring 𝕜'']
[Module 𝕜'' Fₗ]
[ContinuousConstSMul 𝕜'' Fₗ]
[SMulCommClass 𝕜 𝕜'' Fₗ]
[SMulCommClass 𝕜' 𝕜'' Fₗ]
:
(ContinuousLinearMap.restrictScalarsIsometry 𝕜 E Fₗ 𝕜' 𝕜'').toLinearMap = ContinuousLinearMap.restrictScalarsₗ 𝕜 E Fₗ 𝕜' 𝕜''
def
ContinuousLinearMap.restrictScalarsL
(𝕜 : Type u_1)
(E : Type u_4)
(Fₗ : Type u_7)
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
(𝕜' : Type u_11)
[NontriviallyNormedField 𝕜']
[NormedAlgebra 𝕜' 𝕜]
[NormedSpace 𝕜' E]
[IsScalarTower 𝕜' 𝕜 E]
[NormedSpace 𝕜' Fₗ]
[IsScalarTower 𝕜' 𝕜 Fₗ]
(𝕜'' : Type u_12)
[Ring 𝕜'']
[Module 𝕜'' Fₗ]
[ContinuousConstSMul 𝕜'' Fₗ]
[SMulCommClass 𝕜 𝕜'' Fₗ]
[SMulCommClass 𝕜' 𝕜'' Fₗ]
:
ContinuousLinearMap.restrictScalars as a ContinuousLinearMap.
Instances For
@[simp]
theorem
ContinuousLinearMap.coe_restrictScalarsL
{𝕜 : Type u_1}
{E : Type u_4}
{Fₗ : Type u_7}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
{𝕜' : Type u_11}
[NontriviallyNormedField 𝕜']
[NormedAlgebra 𝕜' 𝕜]
[NormedSpace 𝕜' E]
[IsScalarTower 𝕜' 𝕜 E]
[NormedSpace 𝕜' Fₗ]
[IsScalarTower 𝕜' 𝕜 Fₗ]
{𝕜'' : Type u_12}
[Ring 𝕜'']
[Module 𝕜'' Fₗ]
[ContinuousConstSMul 𝕜'' Fₗ]
[SMulCommClass 𝕜 𝕜'' Fₗ]
[SMulCommClass 𝕜' 𝕜'' Fₗ]
:
↑(ContinuousLinearMap.restrictScalarsL 𝕜 E Fₗ 𝕜' 𝕜'') = ContinuousLinearMap.restrictScalarsₗ 𝕜 E Fₗ 𝕜' 𝕜''
@[simp]
theorem
ContinuousLinearMap.coe_restrict_scalarsL'
{𝕜 : Type u_1}
{E : Type u_4}
{Fₗ : Type u_7}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
{𝕜' : Type u_11}
[NontriviallyNormedField 𝕜']
[NormedAlgebra 𝕜' 𝕜]
[NormedSpace 𝕜' E]
[IsScalarTower 𝕜' 𝕜 E]
[NormedSpace 𝕜' Fₗ]
[IsScalarTower 𝕜' 𝕜 Fₗ]
{𝕜'' : Type u_12}
[Ring 𝕜'']
[Module 𝕜'' Fₗ]
[ContinuousConstSMul 𝕜'' Fₗ]
[SMulCommClass 𝕜 𝕜'' Fₗ]
[SMulCommClass 𝕜' 𝕜'' Fₗ]
:
↑(ContinuousLinearMap.restrictScalarsL 𝕜 E Fₗ 𝕜' 𝕜'') = ContinuousLinearMap.restrictScalars 𝕜'
theorem
Submodule.norm_subtypeL_le
{𝕜 : Type u_1}
{E : Type u_4}
[SeminormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
(K : Submodule 𝕜 E)
:
theorem
ContinuousLinearEquiv.lipschitz
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{σ₂₁ : 𝕜₂ →+* 𝕜}
[RingHomInvPair σ₁₂ σ₂₁]
[RingHomInvPair σ₂₁ σ₁₂]
[RingHomIsometric σ₁₂]
(e : E ≃SL[σ₁₂] F)
:
LipschitzWith ‖↑e‖₊ ↑e
theorem
ContinuousLinearEquiv.isBigO_comp
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{σ₂₁ : 𝕜₂ →+* 𝕜}
[RingHomInvPair σ₁₂ σ₂₁]
[RingHomInvPair σ₂₁ σ₁₂]
[RingHomIsometric σ₁₂]
(e : E ≃SL[σ₁₂] F)
{α : Type u_11}
(f : α → E)
(l : Filter α)
:
theorem
ContinuousLinearEquiv.isBigO_sub
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{σ₂₁ : 𝕜₂ →+* 𝕜}
[RingHomInvPair σ₁₂ σ₂₁]
[RingHomInvPair σ₂₁ σ₁₂]
[RingHomIsometric σ₁₂]
(e : E ≃SL[σ₁₂] F)
(l : Filter E)
(x : E)
:
theorem
ContinuousLinearEquiv.isBigO_comp_rev
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{σ₂₁ : 𝕜₂ →+* 𝕜}
[RingHomInvPair σ₁₂ σ₂₁]
[RingHomInvPair σ₂₁ σ₁₂]
[RingHomIsometric σ₂₁]
(e : E ≃SL[σ₁₂] F)
{α : Type u_11}
(f : α → E)
(l : Filter α)
:
theorem
ContinuousLinearEquiv.isBigO_sub_rev
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{σ₂₁ : 𝕜₂ →+* 𝕜}
[RingHomInvPair σ₁₂ σ₂₁]
[RingHomInvPair σ₂₁ σ₁₂]
[RingHomIsometric σ₂₁]
(e : E ≃SL[σ₁₂] F)
(l : Filter E)
(x : E)
:
def
ContinuousLinearMap.bilinearComp
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
{E' : Type u_11}
{F' : Type u_12}
[SeminormedAddCommGroup E']
[SeminormedAddCommGroup F']
{𝕜₁' : Type u_13}
{𝕜₂' : Type u_14}
[NontriviallyNormedField 𝕜₁']
[NontriviallyNormedField 𝕜₂']
[NormedSpace 𝕜₁' E']
[NormedSpace 𝕜₂' F']
{σ₁' : 𝕜₁' →+* 𝕜}
{σ₁₃' : 𝕜₁' →+* 𝕜₃}
{σ₂' : 𝕜₂' →+* 𝕜₂}
{σ₂₃' : 𝕜₂' →+* 𝕜₃}
[RingHomCompTriple σ₁' σ₁₃ σ₁₃']
[RingHomCompTriple σ₂' σ₂₃ σ₂₃']
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃']
[RingHomIsometric σ₂₃']
(f : E →SL[σ₁₃] F →SL[σ₂₃] G)
(gE : E' →SL[σ₁'] E)
(gF : F' →SL[σ₂'] F)
:
Compose a bilinear map E →SL[σ₁₃] F →SL[σ₂₃] G with two linear maps
E' →SL[σ₁'] E and F' →SL[σ₂'] F.
Instances For
@[simp]
theorem
ContinuousLinearMap.bilinearComp_apply
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{𝕜₃ : Type u_3}
{E : Type u_4}
{F : Type u_6}
{G : Type u_8}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup F]
[SeminormedAddCommGroup G]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NontriviallyNormedField 𝕜₃]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
[NormedSpace 𝕜₃ G]
{σ₂₃ : 𝕜₂ →+* 𝕜₃}
{σ₁₃ : 𝕜 →+* 𝕜₃}
{E' : Type u_11}
{F' : Type u_12}
[SeminormedAddCommGroup E']
[SeminormedAddCommGroup F']
{𝕜₁' : Type u_13}
{𝕜₂' : Type u_14}
[NontriviallyNormedField 𝕜₁']
[NontriviallyNormedField 𝕜₂']
[NormedSpace 𝕜₁' E']
[NormedSpace 𝕜₂' F']
{σ₁' : 𝕜₁' →+* 𝕜}
{σ₁₃' : 𝕜₁' →+* 𝕜₃}
{σ₂' : 𝕜₂' →+* 𝕜₂}
{σ₂₃' : 𝕜₂' →+* 𝕜₃}
[RingHomCompTriple σ₁' σ₁₃ σ₁₃']
[RingHomCompTriple σ₂' σ₂₃ σ₂₃']
[RingHomIsometric σ₂₃]
[RingHomIsometric σ₁₃']
[RingHomIsometric σ₂₃']
(f : E →SL[σ₁₃] F →SL[σ₂₃] G)
(gE : E' →SL[σ₁'] E)
(gF : F' →SL[σ₂'] F)
(x : E')
(y : F')
:
↑(↑(ContinuousLinearMap.bilinearComp f gE gF) x) y = ↑(↑f (↑gE x)) (↑gF y)
def
ContinuousLinearMap.deriv₂
{𝕜 : Type u_1}
{E : Type u_4}
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
(f : E →L[𝕜] Fₗ →L[𝕜] Gₗ)
:
Derivative of a continuous bilinear map f : E →L[𝕜] F →L[𝕜] G interpreted as a map E × F → G
at point p : E × F evaluated at q : E × F, as a continuous bilinear map.
Instances For
@[simp]
theorem
ContinuousLinearMap.coe_deriv₂
{𝕜 : Type u_1}
{E : Type u_4}
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
(f : E →L[𝕜] Fₗ →L[𝕜] Gₗ)
(p : E × Fₗ)
:
↑(↑(ContinuousLinearMap.deriv₂ f) p) = fun q => ↑(↑f p.fst) q.snd + ↑(↑f q.fst) p.snd
theorem
ContinuousLinearMap.map_add_add
{𝕜 : Type u_1}
{E : Type u_4}
{Fₗ : Type u_7}
{Gₗ : Type u_9}
[SeminormedAddCommGroup E]
[SeminormedAddCommGroup Fₗ]
[SeminormedAddCommGroup Gₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
[NormedSpace 𝕜 Gₗ]
(f : E →L[𝕜] Fₗ →L[𝕜] Gₗ)
(x : E)
(x' : E)
(y : Fₗ)
(y' : Fₗ)
:
theorem
LinearMap.bound_of_shell
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[NormedAddCommGroup E]
[NormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
(f : E →ₛₗ[σ₁₂] F)
{ε : ℝ}
{C : ℝ}
(ε_pos : 0 < ε)
{c : 𝕜}
(hc : 1 < ‖c‖)
(hf : ∀ (x : E), ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖↑f x‖ ≤ C * ‖x‖)
(x : E)
:
theorem
LinearMap.bound_of_ball_bound
{𝕜 : Type u_1}
{E : Type u_4}
{Fₗ : Type u_7}
[NormedAddCommGroup E]
[NormedAddCommGroup Fₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
{r : ℝ}
(r_pos : 0 < r)
(c : ℝ)
(f : E →ₗ[𝕜] Fₗ)
(h : ∀ (z : E), z ∈ Metric.ball 0 r → ‖↑f z‖ ≤ c)
:
LinearMap.bound_of_ball_bound' is a version of this lemma over a field satisfying IsROrC
that produces a concrete bound.
theorem
LinearMap.antilipschitz_of_comap_nhds_le
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[NormedAddCommGroup E]
[NormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[h : RingHomIsometric σ₁₂]
(f : E →ₛₗ[σ₁₂] F)
(hf : Filter.comap (↑f) (nhds 0) ≤ nhds 0)
:
∃ K, AntilipschitzWith K ↑f
theorem
ContinuousLinearMap.op_norm_zero_iff
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[NormedAddCommGroup E]
[NormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
(f : E →SL[σ₁₂] F)
[RingHomIsometric σ₁₂]
:
An operator is zero iff its norm vanishes.
@[simp]
theorem
ContinuousLinearMap.norm_id
{𝕜 : Type u_1}
{E : Type u_4}
[NormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[Nontrivial E]
:
‖ContinuousLinearMap.id 𝕜 E‖ = 1
If a normed space is non-trivial, then the norm of the identity equals 1.
instance
ContinuousLinearMap.normOneClass
{𝕜 : Type u_1}
{E : Type u_4}
[NormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[Nontrivial E]
:
NormOneClass (E →L[𝕜] E)
instance
ContinuousLinearMap.toNormedAddCommGroup
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[NormedAddCommGroup E]
[NormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
:
NormedAddCommGroup (E →SL[σ₁₂] F)
Continuous linear maps themselves form a normed space with respect to the operator norm.
instance
ContinuousLinearMap.toNormedRing
{𝕜 : Type u_1}
{E : Type u_4}
[NormedAddCommGroup E]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
:
NormedRing (E →L[𝕜] E)
Continuous linear maps form a normed ring with respect to the operator norm.
theorem
ContinuousLinearMap.homothety_norm
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{E : Type u_4}
{F : Type u_6}
[NormedAddCommGroup E]
[NormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
[RingHomIsometric σ₁₂]
[Nontrivial E]
(f : E →SL[σ₁₂] F)
{a : ℝ}
(hf : ∀ (x : E), ‖↑f x‖ = a * ‖x‖)
:
theorem
ContinuousLinearMap.antilipschitz_of_embedding
{𝕜 : Type u_1}
{E : Type u_4}
{Fₗ : Type u_7}
[NormedAddCommGroup E]
[NormedAddCommGroup Fₗ]
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜 Fₗ]
(f : E →L[𝕜] Fₗ)
(hf : Embedding ↑f)
:
∃ K, AntilipschitzWith K ↑f
If a continuous linear map is a topology embedding, then it is expands the distances by a positive factor.
@[simp]
theorem
ContinuousLinearMap.ofMemClosureImageCoeBounded_apply
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{F : Type u_6}
[NormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{E' : Type u_11}
[SeminormedAddCommGroup E']
[NormedSpace 𝕜 E']
[RingHomIsometric σ₁₂]
(f : E' → F)
{s : Set (E' →SL[σ₁₂] F)}
(hs : Bornology.IsBounded s)
(hf : f ∈ closure (FunLike.coe '' s))
:
↑(ContinuousLinearMap.ofMemClosureImageCoeBounded f hs hf) = f
def
ContinuousLinearMap.ofMemClosureImageCoeBounded
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{F : Type u_6}
[NormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{E' : Type u_11}
[SeminormedAddCommGroup E']
[NormedSpace 𝕜 E']
[RingHomIsometric σ₁₂]
(f : E' → F)
{s : Set (E' →SL[σ₁₂] F)}
(hs : Bornology.IsBounded s)
(hf : f ∈ closure (FunLike.coe '' s))
:
Construct a bundled continuous (semi)linear map from a map f : E → F and a proof of the fact
that it belongs to the closure of the image of a bounded set s : Set (E →SL[σ₁₂] F) under coercion
to function. Coercion to function of the result is definitionally equal to f.
Instances For
@[simp]
theorem
ContinuousLinearMap.ofTendstoOfBoundedRange_apply
{𝕜 : Type u_1}
{𝕜₂ : Type u_2}
{F : Type u_6}
[NormedAddCommGroup F]
[NontriviallyNormedField 𝕜]
[NontriviallyNormedField 𝕜₂]
[NormedSpace 𝕜₂ F]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{E' : Type u_11}
[SeminormedAddCommGroup E']
[NormedSpace 𝕜 E']
[RingHomIsometric σ₁₂]
{α : Type u_12}
{l : Filter α}
[Filter.NeBot l]
(f : E' → F)
(g : α → E' →SL[σ₁₂] F)
(hf : Filter.Tendsto (fun a x => ↑(g a) x) l (nhds f))
(hg : Bornology.IsBounded (Set.range g))
:
↑(ContinuousLinearMap.ofTendstoOfBoundedRange f g hf hg) = f