# mathlibdocumentation

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

theorem continuous_linear_map.ker_closed_complemented_of_finite_dimensional_range {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (f : E →L[𝕜] F) [ (f.range)] :
noncomputable def continuous_linear_map.equiv_prod_of_surjective_of_is_compl {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [ E] [ F] [ G] [complete_space (F × G)] (f : E →L[𝕜] F) (g : E →L[𝕜] G) (hf : f.range = ) (hg : g.range = ) (hfg : 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
• hg hfg =
@[simp]
theorem continuous_linear_map.coe_equiv_prod_of_surjective_of_is_compl {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [ E] [ F] [ G] [complete_space (F × G)] {f : E →L[𝕜] F} {g : E →L[𝕜] G} (hf : f.range = ) (hg : g.range = ) (hfg : g.ker) :
hg hfg) = (f.prod g)
@[simp]
theorem continuous_linear_map.equiv_prod_of_surjective_of_is_compl_to_linear_equiv {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [ E] [ F] [ G] [complete_space (F × G)] {f : E →L[𝕜] F} {g : E →L[𝕜] G} (hf : f.range = ) (hg : g.range = ) (hfg : g.ker) :
hg hfg).to_linear_equiv = hfg
@[simp]
theorem continuous_linear_map.equiv_prod_of_surjective_of_is_compl_apply {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [ E] [ F] [ G] [complete_space (F × G)] {f : E →L[𝕜] F} {g : E →L[𝕜] G} (hf : f.range = ) (hg : g.range = ) (hfg : g.ker) (x : E) :
hg hfg) x = (f x, g x)
noncomputable def subspace.prod_equiv_of_closed_compl {𝕜 : Type u_1} {E : Type u_2} [ E] (p q : E) (h : q) (hp : is_closed p) (hq : is_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
noncomputable def subspace.linear_proj_of_closed_compl {𝕜 : Type u_1} {E : Type u_2} [ E] (p q : E) (h : q) (hp : is_closed p) (hq : is_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} {E : Type u_2} [ E] {p q : E} (h : q) (hp : is_closed p) (hq : is_closed q) :
hp hq) =
@[simp]
theorem subspace.coe_prod_equiv_of_closed_compl_symm {𝕜 : Type u_1} {E : Type u_2} [ E] {p q : E} (h : q) (hp : is_closed p) (hq : is_closed q) :
h hp hq).symm) = h).symm)
@[simp]
theorem subspace.coe_continuous_linear_proj_of_closed_compl {𝕜 : Type u_1} {E : Type u_2} [ E] {p q : E} (h : q) (hp : is_closed p) (hq : is_closed q) :
hp hq) =
@[simp]
theorem subspace.coe_continuous_linear_proj_of_closed_compl' {𝕜 : Type u_1} {E : Type u_2} [ E] {p q : E} (h : q) (hp : is_closed p) (hq : is_closed q) :
hp hq) =
theorem subspace.closed_complemented_of_closed_compl {𝕜 : Type u_1} {E : Type u_2} [ E] {p q : E} (h : q) (hp : is_closed p) (hq : is_closed q) :
theorem subspace.closed_complemented_iff_has_closed_compl {𝕜 : Type u_1} {E : Type u_2} [ E] {p : E} :
∃ (q : E) (hq : , q
theorem subspace.closed_complemented_of_quotient_finite_dimensional {𝕜 : Type u_1} {E : Type u_2} [ E] {p : E} [ (E p)] (hp : is_closed p) :