mathlib documentation

analysis.normed_space.complemented

Complemented subspaces of normed vector spaces

A submodule p of a topological module E over R is called complemented if there exists a continuous linear projection f : E →ₗ[R] p, ∀ x : p, f x = x. We prove that for a closed subspace of a normed space this condition is equivalent to existence of a closed subspace q such that p ⊓ q = ⊥, p ⊔ q = ⊤. We also prove that a subspace of finite codimension is always a complemented subspace.

Tags

complemented subspace, normed vector space

def continuous_linear_map.equiv_prod_of_surjective_of_is_compl {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] [complete_space E] [complete_space (F × G)] (f : E →L[𝕜] F) (g : E →L[𝕜] G) :
f.range = g.range = is_compl f.ker g.ker(E ≃L[𝕜] F × G)

If f : E →L[R] F and g : E →L[R] G are two surjective linear maps and their kernels are complement of each other, then x ↦ (f x, g x) defines a linear equivalence E ≃L[R] F × G.

Equations
@[simp]
theorem continuous_linear_map.coe_equiv_prod_of_surjective_of_is_compl {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] [complete_space E] [complete_space (F × G)] {f : E →L[𝕜] F} {g : E →L[𝕜] G} (hf : f.range = ) (hg : g.range = ) (hfg : is_compl f.ker g.ker) :

@[simp]
theorem continuous_linear_map.equiv_prod_of_surjective_of_is_compl_to_linear_equiv {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] [complete_space E] [complete_space (F × G)] {f : E →L[𝕜] F} {g : E →L[𝕜] G} (hf : f.range = ) (hg : g.range = ) (hfg : is_compl f.ker g.ker) :

@[simp]
theorem continuous_linear_map.equiv_prod_of_surjective_of_is_compl_apply {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] [complete_space E] [complete_space (F × G)] {f : E →L[𝕜] F} {g : E →L[𝕜] G} (hf : f.range = ) (hg : g.range = ) (hfg : is_compl f.ker g.ker) (x : E) :

def subspace.prod_equiv_of_closed_compl {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] [complete_space E] (p q : subspace 𝕜 E) :
is_compl p qis_closed pis_closed q((p × q) ≃L[𝕜] E)

If q is a closed complement of a closed subspace p, then p × q is continuously isomorphic to E.

Equations
def subspace.linear_proj_of_closed_compl {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] [complete_space E] (p q : subspace 𝕜 E) :
is_compl p qis_closed pis_closed q(E →L[𝕜] p)

Projection to a closed submodule along a closed complement.

Equations
@[simp]
theorem subspace.coe_prod_equiv_of_closed_compl {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] [complete_space E] {p q : subspace 𝕜 E} (h : is_compl p q) (hp : is_closed p) (hq : is_closed q) :

@[simp]
theorem subspace.coe_prod_equiv_of_closed_compl_symm {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] [complete_space E] {p q : subspace 𝕜 E} (h : is_compl p q) (hp : is_closed p) (hq : is_closed q) :

@[simp]

@[simp]