# mathlibdocumentation

analysis.normed_space.banach

# Banach open mapping theorem #

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 {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] 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 β₯inverse xβ₯ β€ C * β₯xβ₯. 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 {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) :
(Ξ» (_x : f.nonlinear_right_inverse), F β E)
Equations
@[simp]
theorem continuous_linear_map.nonlinear_right_inverse.right_inv {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {f : E βL[π] F} (fsymm : f.nonlinear_right_inverse) (y : F) :
βf (βfsymm y) = y
theorem continuous_linear_map.nonlinear_right_inverse.bound {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {f : E βL[π] F} (fsymm : f.nonlinear_right_inverse) (y : F) :
noncomputable def continuous_linear_equiv.to_nonlinear_right_inverse {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] 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 {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) :
Equations

### Proof of the Banach open mapping theorem #

theorem continuous_linear_map.exists_approx_preimage_norm_le {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (surj : function.surjective βf) :
β (C : β) (H : C β₯ 0), β (y : F), β (x : E), has_dist.dist (βf x) y β€ 1 / 2 * β₯yβ₯ β§ β₯xβ₯ β€ C * β₯yβ₯

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 β F is arbitrarily well approached by images of elements of norm at most C * β₯yβ₯. For further use, we will only need such an element whose image is within distance β₯yβ₯/2 of y, to apply an iterative process.

theorem continuous_linear_map.exists_preimage_norm_le {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (surj : function.surjective βf) :
β (C : β) (H : C > 0), β (y : F), β (x : E), βf x = y β§ β₯xβ₯ β€ C * β₯yβ₯

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 {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (surj : function.surjective βf) :

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

@[protected]
theorem continuous_linear_map.quotient_map {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (surj : function.surjective βf) :
theorem affine_map.is_open_map {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {P : Type u_4} {Q : Type u_5} [metric_space P] [ P] [metric_space Q] [ Q] (f : P βα΅[π] Q) (hf : continuous βf) (surj : function.surjective βf) :

### Applications of the Banach open mapping theorem #

theorem continuous_linear_map.interior_preimage {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (hsurj : function.surjective βf) (s : set F) :
theorem continuous_linear_map.closure_preimage {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (hsurj : function.surjective βf) (s : set F) :
theorem continuous_linear_map.frontier_preimage {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (hsurj : function.surjective βf) (s : set F) :
theorem continuous_linear_map.exists_nonlinear_right_inverse_of_surjective {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (hsurj : f.range = β€) :
β (fsymm : f.nonlinear_right_inverse), 0 < fsymm.nnnorm
@[irreducible]
noncomputable def continuous_linear_map.nonlinear_right_inverse_of_surjective {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (hsurj : f.range = β€) :

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 {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (hsurj : f.range = β€) :
@[continuity]
theorem linear_equiv.continuous_symm {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (e : E ββ[π] F) (h : continuous β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 {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (e : E ββ[π] F) (h : continuous βe) :
E βL[π] 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 {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (e : E ββ[π] F) (h : continuous βe) :
@[simp]
theorem linear_equiv.coe_fn_to_continuous_linear_equiv_of_continuous_symm {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (e : E ββ[π] F) (h : continuous βe) :
noncomputable def continuous_linear_equiv.of_bijective {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (hinj : f.ker = β₯) (hsurj : f.range = β€) :
E βL[π] F

Convert a bijective continuous linear map f : E βL[π] F from a Banach space to a normed space to a continuous linear equivalence.

Equations
@[simp]
theorem continuous_linear_equiv.coe_fn_of_bijective {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (hinj : f.ker = β₯) (hsurj : f.range = β€) :
β hsurj) = βf
theorem continuous_linear_equiv.coe_of_bijective {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (hinj : f.ker = β₯) (hsurj : f.range = β€) :
β hsurj) = f
@[simp]
theorem continuous_linear_equiv.of_bijective_symm_apply_apply {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (hinj : f.ker = β₯) (hsurj : f.range = β€) (x : E) :
β hsurj).symm) (βf x) = x
@[simp]
theorem continuous_linear_equiv.of_bijective_apply_symm_apply {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (hinj : f.ker = β₯) (hsurj : f.range = β€) (y : F) :
βf (β hsurj).symm) y) = y
noncomputable def continuous_linear_map.coprod_subtypeL_equiv_of_is_compl {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) {G : submodule π F} (h : G) (hker : f.ker = β₯) :
(E Γ β₯G) βL[π] 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 {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) {G : submodule π F} (h : G) (hker : f.ker = β₯) :
theorem continuous_linear_map.closed_complemented_range_of_is_compl_of_ker_eq_bot {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (G : submodule π F) (h : G) (hG : is_closed βG) (hker : f.ker = β₯) :
theorem linear_map.continuous_of_is_closed_graph {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (g : E ββ[π] F) (hg : is_closed β(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 {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (g : E ββ[π] F) (hg : β (u : β β E) (x : E) (y : F), β (nhds y) β y = β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 uβ βΆ x, if f(uβ) βΆ y then y = f(x).

def continuous_linear_map.of_is_closed_graph {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {g : E ββ[π] F} (hg : is_closed β(g.graph)) :
E βL[π] 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 {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {g : E ββ[π] F} (hg : is_closed β(g.graph)) :
theorem continuous_linear_map.coe_of_is_closed_graph {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {g : E ββ[π] F} (hg : is_closed β(g.graph)) :
def continuous_linear_map.of_seq_closed_graph {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {g : E ββ[π] F} (hg : β (u : β β E) (x : E) (y : F), β (nhds y) β y = βg x) :
E βL[π] 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 {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {g : E ββ[π] F} (hg : β (u : β β E) (x : E) (y : F), β (nhds y) β y = βg x) :
theorem continuous_linear_map.coe_of_seq_closed_graph {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {g : E ββ[π] F} (hg : β (u : β β E) (x : E) (y : F), β (nhds y) β y = βg x) :