Extension Hahn-Banach theorem

In this file we prove the analytic Hahn-Banach theorem. For any continuous linear function on a subspace, we can extend it to a function on the entire space without changing its norm.

We prove

• Real.exists_extension_norm_eq: Hahn-Banach theorem for continuous linear functions on normed spaces over ℝ.
• exists_extension_norm_eq: Hahn-Banach theorem for continuous linear functions on normed spaces over ℝ or ℂ.

In order to state and prove the corollaries uniformly, we prove the statements for a field 𝕜 satisfying RCLike 𝕜.

In this setting, exists_dual_vector states that, for any nonzero x, there exists a continuous linear form g of norm 1 with g x = ‖x‖ (where the norm has to be interpreted as an element of 𝕜).

theorem Real.exists_extension_norm_eq {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (p : Subspace ℝ E) (f : ↥p →L[ℝ] ℝ) : ∃ (g : E →L[ℝ] ℝ), (∀ (x : ↥p), g ↑x = f x) ∧ ‖g‖ = ‖f‖

Hahn-Banach theorem for continuous linear functions over ℝ. See also exists_extension_norm_eq in the root namespace for a more general version that works both for ℝ and ℂ.

theorem exists_extension_norm_eq {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (p : Subspace 𝕜 E) (f : ↥p →L[𝕜] 𝕜) : ∃ (g : E →L[𝕜] 𝕜), (∀ (x : ↥p), g ↑x = f x) ∧ ‖g‖ = ‖f‖

Hahn-Banach theorem for continuous linear functions over 𝕜 satisfying RCLike 𝕜.

theorem ContinuousLinearMap.exist_extension_of_finiteDimensional_range {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} {F : Type u_3} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : Submodule 𝕜 E} (f : ↥p →L[𝕜] F) [FiniteDimensional 𝕜 ↥(LinearMap.range f)] : ∃ (g : E →L[𝕜] F), f = ContinuousLinearMap.comp g (Submodule.subtypeL p)

Corollary of the Hahn-Banach theorem: if f : p → F is a continuous linear map from a submodule of a normed space E over 𝕜, 𝕜 = ℝ or 𝕜 = ℂ, with a finite dimensional range, then f admits an extension to a continuous linear map E → F.

Note that contrary to the case F = 𝕜, see exists_extension_norm_eq, we provide no estimates on the norm of the extension.

theorem Submodule.ClosedComplemented.of_finiteDimensional {𝕜 : Type u_1} [RCLike 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : Submodule 𝕜 F) [FiniteDimensional 𝕜 ↥p] : Submodule.ClosedComplemented p

A finite dimensional submodule over ℝ or ℂ is Submodule.ClosedComplemented.

theorem coord_norm' (𝕜 : Type v) [RCLike 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} (h : x ≠ 0) : ‖↑‖x‖ • ContinuousLinearEquiv.coord 𝕜 x h‖ = 1

theorem exists_dual_vector (𝕜 : Type v) [RCLike 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E) (h : x ≠ 0) : ∃ (g : E →L[𝕜] 𝕜), ‖g‖ = 1 ∧ g x = ↑‖x‖

Corollary of Hahn-Banach. Given a nonzero element x of a normed space, there exists an element of the dual space, of norm 1, whose value on x is ‖x‖.

theorem exists_dual_vector' (𝕜 : Type v) [RCLike 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [Nontrivial E] (x : E) : ∃ (g : E →L[𝕜] 𝕜), ‖g‖ = 1 ∧ g x = ↑‖x‖

Variant of Hahn-Banach, eliminating the hypothesis that x be nonzero, and choosing the dual element arbitrarily when x = 0.

theorem exists_dual_vector'' (𝕜 : Type v) [RCLike 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E) : ∃ (g : E →L[𝕜] 𝕜), ‖g‖ ≤ 1 ∧ g x = ↑‖x‖

Variant of Hahn-Banach, eliminating the hypothesis that x be nonzero, but only ensuring that the dual element has norm at most 1 (this can not be improved for the trivial vector space).