Extension Hahn-Banach theorem #

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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 is_R_or_C 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).

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theorem real.exists_extension_norm_eq {E : Type u_1} [seminormed_add_comm_group E] [normed_space 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.

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theorem exists_extension_norm_eq {𝕜 : Type u_1} [is_R_or_C 𝕜] {F : Type u_2} [seminormed_add_comm_group F] [normed_space 𝕜 F] (p : subspace 𝕜 F) (f : ↥p →L[𝕜] 𝕜) :

∃ (g : F →L[𝕜] 𝕜), (∀ (x : ↥p), ⇑g ↑x = ⇑f x) ∧ ‖g‖ = ‖f‖

Hahn-Banach theorem for continuous linear functions over 𝕜 satisyfing is_R_or_C 𝕜.

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theorem coord_norm' (𝕜 : Type v) [is_R_or_C 𝕜] {E : Type u} [normed_add_comm_group E] [normed_space 𝕜 E] {x : E} (h : x ≠ 0) :

‖↑‖x‖ • continuous_linear_equiv.coord 𝕜 x h‖ = 1

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theorem exists_dual_vector (𝕜 : Type v) [is_R_or_C 𝕜] {E : Type u} [normed_add_comm_group E] [normed_space 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‖.

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theorem exists_dual_vector' (𝕜 : Type v) [is_R_or_C 𝕜] {E : Type u} [normed_add_comm_group E] [normed_space 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.

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theorem exists_dual_vector'' (𝕜 : Type v) [is_R_or_C 𝕜] {E : Type u} [normed_add_comm_group E] [normed_space 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).