# mathlib3documentation

analysis.normed_space.banach

# Banach open mapping theorem #

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This file contains the Banach open mapping theorem, i.e., the fact that a bijective bounded linear map between Banach spaces has a bounded inverse.

structure continuous_linear_map.nonlinear_right_inverse {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) :
Type (max u_2 u_3)

A (possibly nonlinear) right inverse to a continuous linear map, which doesn't have to be linear itself but which satisfies a bound ‖inverse x‖ ≤ C * ‖x‖. A surjective continuous linear map doesn't always have a continuous linear right inverse, but it always has a nonlinear inverse in this sense, by Banach's open mapping theorem.

Instances for continuous_linear_map.nonlinear_right_inverse
@[protected, instance]
def continuous_linear_map.nonlinear_right_inverse.has_coe_to_fun {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) :
(λ (_x : f.nonlinear_right_inverse), F E)
Equations
@[simp]
theorem continuous_linear_map.nonlinear_right_inverse.right_inv {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] {f : E →L[𝕜] F} (fsymm : f.nonlinear_right_inverse) (y : F) :
f (fsymm y) = y
theorem continuous_linear_map.nonlinear_right_inverse.bound {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] {f : E →L[𝕜] F} (fsymm : f.nonlinear_right_inverse) (y : F) :
fsymm y (fsymm.nnnorm) * y
noncomputable def continuous_linear_equiv.to_nonlinear_right_inverse {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E ≃L[𝕜] F) :

Given a continuous linear equivalence, the inverse is in particular an instance of nonlinear_right_inverse (which turns out to be linear).

Equations
@[protected, instance]
noncomputable def continuous_linear_map.nonlinear_right_inverse.inhabited {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E ≃L[𝕜] F) :
Equations

### Proof of the Banach open mapping theorem #

theorem continuous_linear_map.exists_approx_preimage_norm_le {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (surj : function.surjective f) :
(C : ) (H : C 0), (y : F), (x : E), has_dist.dist (f x) y 1 / 2 * y x C * y

First step of the proof of the Banach open mapping theorem (using completeness of F): by Baire's theorem, there exists a ball in E whose image closure has nonempty interior. Rescaling everything, it follows that any y ∈ F is arbitrarily well approached by images of elements of norm at most C * ‖y‖. For further use, we will only need such an element whose image is within distance ‖y‖/2 of y, to apply an iterative process.

theorem continuous_linear_map.exists_preimage_norm_le {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (surj : function.surjective f) :
(C : ) (H : C > 0), (y : F), (x : E), f x = y x C * y

The Banach open mapping theorem: if a bounded linear map between Banach spaces is onto, then any point has a preimage with controlled norm.

@[protected]
theorem continuous_linear_map.is_open_map {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (surj : function.surjective f) :

The Banach open mapping theorem: a surjective bounded linear map between Banach spaces is open.

@[protected]
theorem continuous_linear_map.quotient_map {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (surj : function.surjective f) :
theorem affine_map.is_open_map {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] {P : Type u_4} {Q : Type u_5} [metric_space P] [ P] [metric_space Q] [ Q] (f : P →ᵃ[𝕜] Q) (hf : continuous f) (surj : function.surjective f) :

### Applications of the Banach open mapping theorem #

theorem continuous_linear_map.interior_preimage {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (hsurj : function.surjective f) (s : set F) :
theorem continuous_linear_map.closure_preimage {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (hsurj : function.surjective f) (s : set F) :
theorem continuous_linear_map.frontier_preimage {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (hsurj : function.surjective f) (s : set F) :
theorem continuous_linear_map.exists_nonlinear_right_inverse_of_surjective {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (hsurj : = ) :
(fsymm : f.nonlinear_right_inverse), 0 < fsymm.nnnorm
@[irreducible]
noncomputable def continuous_linear_map.nonlinear_right_inverse_of_surjective {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (hsurj : = ) :

A surjective continuous linear map between Banach spaces admits a (possibly nonlinear) controlled right inverse. In general, it is not possible to ensure that such a right inverse is linear (take for instance the map from E to E/F where F is a closed subspace of E without a closed complement. Then it doesn't have a continuous linear right inverse.)

Equations
theorem continuous_linear_map.nonlinear_right_inverse_of_surjective_nnnorm_pos {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (hsurj : = ) :
@[continuity]
theorem linear_equiv.continuous_symm {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (e : E ≃ₗ[𝕜] F) (h : continuous e) :

If a bounded linear map is a bijection, then its inverse is also a bounded linear map.

def linear_equiv.to_continuous_linear_equiv_of_continuous {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (e : E ≃ₗ[𝕜] F) (h : continuous e) :
E ≃L[𝕜] F

Associating to a linear equivalence between Banach spaces a continuous linear equivalence when the direct map is continuous, thanks to the Banach open mapping theorem that ensures that the inverse map is also continuous.

Equations
@[simp]
theorem linear_equiv.coe_fn_to_continuous_linear_equiv_of_continuous {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (e : E ≃ₗ[𝕜] F) (h : continuous e) :
@[simp]
theorem linear_equiv.coe_fn_to_continuous_linear_equiv_of_continuous_symm {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (e : E ≃ₗ[𝕜] F) (h : continuous e) :
= (e.symm)
noncomputable def continuous_linear_equiv.of_bijective {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (hinj : = ) (hsurj : = ) :
E ≃L[𝕜] F

Convert a bijective continuous linear map f : E →L[𝕜] F from a Banach space to a normed space to a continuous linear equivalence.

Equations
@[simp]
theorem continuous_linear_equiv.coe_fn_of_bijective {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (hinj : = ) (hsurj : = ) :
hsurj) = f
theorem continuous_linear_equiv.coe_of_bijective {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (hinj : = ) (hsurj : = ) :
hsurj) = f
@[simp]
theorem continuous_linear_equiv.of_bijective_symm_apply_apply {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (hinj : = ) (hsurj : = ) (x : E) :
hsurj).symm) (f x) = x
@[simp]
theorem continuous_linear_equiv.of_bijective_apply_symm_apply {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (hinj : = ) (hsurj : = ) (y : F) :
f ( hsurj).symm) y) = y
noncomputable def continuous_linear_map.coprod_subtypeL_equiv_of_is_compl {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) {G : F} (h : G) (hker : = ) :
(E × G) ≃L[𝕜] F

Intermediate definition used to show continuous_linear_map.closed_complemented_range_of_is_compl_of_ker_eq_bot.

This is f.coprod G.subtypeL as an continuous_linear_equiv.

Equations
theorem continuous_linear_map.range_eq_map_coprod_subtypeL_equiv_of_is_compl {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) {G : F} (h : G) (hker : = ) :
theorem continuous_linear_map.closed_complemented_range_of_is_compl_of_ker_eq_bot {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (f : E →L[𝕜] F) (G : F) (h : G) (hG : is_closed G) (hker : = ) :
theorem linear_map.continuous_of_is_closed_graph {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (g : E →ₗ[𝕜] F) (hg : is_closed (g.graph)) :

The closed graph theorem : a linear map between two Banach spaces whose graph is closed is continuous.

theorem linear_map.continuous_of_seq_closed_graph {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] (g : E →ₗ[𝕜] F) (hg : (u : E) (x : E) (y : F), (nhds y) y = g x) :

A useful form of the closed graph theorem : let f be a linear map between two Banach spaces. To show that f is continuous, it suffices to show that for any convergent sequence uₙ ⟶ x, if f(uₙ) ⟶ y then y = f(x).

def continuous_linear_map.of_is_closed_graph {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] {g : E →ₗ[𝕜] F} (hg : is_closed (g.graph)) :
E →L[𝕜] F

Upgrade a linear_map to a continuous_linear_map using the closed graph theorem.

Equations
@[simp]
theorem continuous_linear_map.coe_fn_of_is_closed_graph {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] {g : E →ₗ[𝕜] F} (hg : is_closed (g.graph)) :
theorem continuous_linear_map.coe_of_is_closed_graph {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] {g : E →ₗ[𝕜] F} (hg : is_closed (g.graph)) :
def continuous_linear_map.of_seq_closed_graph {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] {g : E →ₗ[𝕜] F} (hg : (u : E) (x : E) (y : F), (nhds y) y = g x) :
E →L[𝕜] F

Upgrade a linear_map to a continuous_linear_map using a variation on the closed graph theorem.

Equations
@[simp]
theorem continuous_linear_map.coe_fn_of_seq_closed_graph {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] {g : E →ₗ[𝕜] F} (hg : (u : E) (x : E) (y : F), (nhds y) y = g x) :
theorem continuous_linear_map.coe_of_seq_closed_graph {𝕜 : Type u_1} {E : Type u_2} [ E] {F : Type u_3} [ F] {g : E →ₗ[𝕜] F} (hg : (u : E) (x : E) (y : F), (nhds y) y = g x) :