# mathlib3documentation

analysis.normed_space.hahn_banach.separation

# Separation Hahn-Banach theorem #

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In this file we prove the geometric Hahn-Banach theorem. For any two disjoint convex sets, there exists a continuous linear functional separating them, geometrically meaning that we can intercalate a plane between them.

We provide many variations to stricten the result under more assumptions on the convex sets:

• geometric_hahn_banach_open: One set is open. Weak separation.
• geometric_hahn_banach_open_point, geometric_hahn_banach_point_open: One set is open, the other is a singleton. Weak separation.
• geometric_hahn_banach_open_open: Both sets are open. Semistrict separation.
• geometric_hahn_banach_compact_closed, geometric_hahn_banach_closed_compact: One set is closed, the other one is compact. Strict separation.
• geometric_hahn_banach_point_closed, geometric_hahn_banach_closed_point: One set is closed, the other one is a singleton. Strict separation.
• geometric_hahn_banach_point_point: Both sets are singletons. Strict separation.

## TODO #

• Eidelheit's theorem
• convex ℝ s → interior (closure s) ⊆ s
theorem separate_convex_open_set {E : Type u_2} [ E] {s : set E} (hs₀ : 0 s) (hs₁ : s) (hs₂ : is_open s) {x₀ : E} (hx₀ : x₀ s) :
(f : E →L[] ), f x₀ = 1 (x : E), x s f x < 1

Given a set s which is a convex neighbourhood of 0 and a point x₀ outside of it, there is a continuous linear functional f separating x₀ and s, in the sense that it sends x₀ to 1 and all of s to values strictly below 1.

theorem geometric_hahn_banach_open {E : Type u_2} [ E] {s t : set E} (hs₁ : s) (hs₂ : is_open s) (ht : t) (disj : t) :
(f : E →L[] ) (u : ), ( (a : E), a s f a < u) (b : E), b t u f b

A version of the Hahn-Banach theorem: given disjoint convex sets s, t where s is open, there is a continuous linear functional which separates them.

theorem geometric_hahn_banach_open_point {E : Type u_2} [ E] {s : set E} {x : E} (hs₁ : s) (hs₂ : is_open s) (disj : x s) :
(f : E →L[] ), (a : E), a s f a < f x
theorem geometric_hahn_banach_point_open {E : Type u_2} [ E] {t : set E} {x : E} (ht₁ : t) (ht₂ : is_open t) (disj : x t) :
(f : E →L[] ), (b : E), b t f x < f b
theorem geometric_hahn_banach_open_open {E : Type u_2} [ E] {s t : set E} (hs₁ : s) (hs₂ : is_open s) (ht₁ : t) (ht₃ : is_open t) (disj : t) :
(f : E →L[] ) (u : ), ( (a : E), a s f a < u) (b : E), b t u < f b
theorem geometric_hahn_banach_compact_closed {E : Type u_2} [ E] {s t : set E} (hs₁ : s) (hs₂ : is_compact s) (ht₁ : t) (ht₂ : is_closed t) (disj : t) :
(f : E →L[] ) (u v : ), ( (a : E), a s f a < u) u < v (b : E), b t v < f b

A version of the Hahn-Banach theorem: given disjoint convex sets s, t where s is compact and t is closed, there is a continuous linear functional which strongly separates them.

theorem geometric_hahn_banach_closed_compact {E : Type u_2} [ E] {s t : set E} (hs₁ : s) (hs₂ : is_closed s) (ht₁ : t) (ht₂ : is_compact t) (disj : t) :
(f : E →L[] ) (u v : ), ( (a : E), a s f a < u) u < v (b : E), b t v < f b

A version of the Hahn-Banach theorem: given disjoint convex sets s, t where s is closed, and t is compact, there is a continuous linear functional which strongly separates them.

theorem geometric_hahn_banach_point_closed {E : Type u_2} [ E] {t : set E} {x : E} (ht₁ : t) (ht₂ : is_closed t) (disj : x t) :
(f : E →L[] ) (u : ), f x < u (b : E), b t u < f b
theorem geometric_hahn_banach_closed_point {E : Type u_2} [ E] {s : set E} {x : E} (hs₁ : s) (hs₂ : is_closed s) (disj : x s) :
(f : E →L[] ) (u : ), ( (a : E), a s f a < u) u < f x
theorem geometric_hahn_banach_point_point {E : Type u_2} [ E] {x y : E} [t1_space E] (hxy : x y) :
(f : E →L[] ), f x < f y

See also normed_space.eq_iff_forall_dual_eq.

theorem Inter_halfspaces_eq {E : Type u_2} [ E] {s : set E} (hs₁ : s) (hs₂ : is_closed s) :
( (l : E →L[] ), {x : E | (y : E) (H : y s), l x l y}) = s

A closed convex set is the intersection of the halfspaces containing it.