Editing Hahn–Banach theorem (section)

==Geometric Hahn–Banach (the Hahn–Banach separation theorems)==
{{See also|Hyperplane separation theorem}}

The key element of the Hahn–Banach theorem is fundamentally a result about the separation of two convex sets: <math>\{-p(- x - n) - f(n) : n \in M\},</math> and <math>\{p(m + x) - f(m) : m \in M\}.</math>  This sort of argument appears widely in [[convex geometry]],<ref>{{cite journal|last1=Harvey|first1=R.|last2=Lawson|first2=H. B.|year=1983|title=An intrinsic characterisation of Kähler manifolds|journal=[[Inventiones Mathematicae|Invent. Math.]]|volume=74|issue=2|pages=169–198|doi=10.1007/BF01394312|bibcode=1983InMat..74..169H|s2cid=124399104}}</ref> [[optimization (mathematics)|optimization theory]], and [[mathematical economics#Functional analysis|economics]].  Lemmas to this end derived from the original Hahn–Banach theorem are known as the '''Hahn–Banach separation theorems'''.<ref name="Zalinescu">{{cite book|last=Zălinescu|first=C.|title=Convex analysis in general vector spaces|publisher=World Scientific Publishing Co., Inc|location= River Edge, NJ |date= 2002|pages=5–7|isbn=981-238-067-1|mr=1921556}}</ref><ref>Gabriel Nagy, [http://www.math.ksu.edu/~nagy/real-an/ap-e-h-b.pdf Real Analysis] [http://www.math.ksu.edu/~nagy/real-an/ lecture notes]</ref> 
They are generalizations of the [[hyperplane separation theorem]], which states that two disjoint nonempty convex subsets of a finite-dimensional space <math>\R^n</math> can be separated by some {{em|[[affine hyperplane]]}}, which is a [[Fiber (mathematics)|fiber]] ([[level set]]) of the form <math>f^{-1}(s) = \{x : f(x) = s\}</math> where <math>f \neq 0</math> is a non-zero linear functional and <math>s</math> is a scalar. 

{{Math theorem
| name = Theorem<ref name="Zalinescu"/>
| math_statement = Let <math>A</math> and <math>B</math> be non-empty convex subsets of a real [[locally convex topological vector space]] <math>X.</math> 
If <math>\operatorname{Int} A \neq \varnothing</math> and <math>B \cap \operatorname{Int} A = \varnothing</math> then there exists a continuous linear functional <math>f</math> on <math>X</math> such that <math>\sup f(A) \leq \inf f(B)</math> and <math>f(a) < \inf f(B)</math> for all <math>a \in \operatorname{Int} A</math> (such an <math>f</math> is necessarily non-zero).
}}

When the convex sets have additional properties, such as being [[Open set|open]] or [[Compact set|compact]] for example, then the conclusion can be substantially strengthened:

{{Math theorem
| name = Theorem{{sfn|Narici|Beckenstein|2011|pp=177-220}}<ref>{{cite book|last=Brezis|first=Haim|publisher=Springer|location=New York|date=2011|pages=6–7|title=Functional Analysis, Sobolev Spaces, and Partial Differential Equations}}</ref>
| math_statement = Let <math>A</math> and <math>B</math> be convex non-empty disjoint subsets of a real [[topological vector space]] <math>X.</math>  
* If <math>A</math> is open then <math>A</math> and <math>B</math> are ''separated by a closed hyperplane''.  Explicitly, this means that there exists a continuous linear map <math>f : X \to \mathbf{K}</math> and <math>s \in \R</math> such that <math>f(a) < s \leq f(b)</math> for all <math>a \in A, b \in B.</math>  If both <math>A</math> and <math>B</math> are open then the right-hand side may be taken strict as well.
* If <math>X</math> is locally convex, <math>A</math> is compact, and <math>B</math> closed, then <math>A</math> and <math>B</math> are ''strictly separated'': there exists a continuous linear map <math>f : X \to \mathbf{K}</math> and <math>s, t \in \R</math> such that <math>f(a) < t <  s < f(b)</math> for all <math>a \in A, b \in B.</math>

If <math>X</math> is complex (rather than real) then the same claims hold, but for the [[real part]] of <math>f.</math>
}}

Then following important corollary is known as the '''Geometric Hahn–Banach theorem''' or '''Mazur's theorem''' (also known as '''Ascoli–Mazur theorem'''<ref>{{cite book|first1=Semen|last1=Kutateladze|date=1996|pages=40|title=Fundamentals of Functional Analysis|series=Kluwer Texts in the Mathematical Sciences |volume=12|isbn=978-90-481-4661-1|url=https://www.researchgate.net/publication/240011075|doi=10.1007/978-94-015-8755-6}}</ref>). It follows from the first bullet above and the convexity of <math>M.</math> 

{{Math theorem
| name = Theorem (Mazur){{sfn|Trèves|2006|p=184}}
| math_statement = Let <math>M</math> be a vector subspace of the topological vector space <math>X</math> and suppose <math>K</math> is a non-empty convex open subset of <math>X</math> with <math>K \cap M = \varnothing.</math> 
Then there is a closed [[hyperplane]] (codimension-1 vector subspace) <math>N \subseteq X</math> that contains <math>M,</math> but remains disjoint from <math>K.</math>
}}

Mazur's theorem clarifies that vector subspaces (even those that are not closed) can be characterized by linear functionals. 

{{Math theorem
| name = Corollary{{sfn|Narici|Beckenstein|2011|pp=195}}
| note = Separation of a subspace and an open convex set
| math_statement = Let <math>M</math> be a vector subspace of a [[locally convex topological vector space]] <math>X,</math> and <math>U</math> be a non-empty open convex subset disjoint from <math>M.</math>  Then there exists a continuous linear functional <math>f</math> on <math>X</math> such that <math>f(m) = 0</math> for all <math>m \in M</math> and <math>\operatorname{Re} f > 0</math> on <math>U.</math>
}}

===Supporting hyperplanes===

Since points are trivially [[Convex set|convex]], geometric Hahn–Banach implies that functionals can detect the [[Boundary (topology)|boundary]] of a set.  In particular, let <math>X</math> be a real topological vector space and <math>A \subseteq X</math> be convex with <math>\operatorname{Int} A \neq \varnothing.</math> If <math>a_0 \in A \setminus \operatorname{Int} A</math> then there is a functional that is vanishing at <math>a_0,</math> but supported on the interior of <math>A.</math><ref name="Zalinescu" />

Call a normed space <math>X</math> '''smooth''' if at each point <math>x</math> in its unit ball there exists a unique closed hyperplane to the unit ball at <math>x.</math> Köthe showed in 1983 that a normed space is smooth at a point <math>x</math> if and only if the norm is [[Gateaux derivative|Gateaux differentiable]] at that point.{{sfn|Narici|Beckenstein|2011|pp=177-220}}

===Balanced or disked neighborhoods===

Let <math>U</math> be a convex [[Balanced set|balanced]] neighborhood of the origin in a [[locally convex]] topological vector space <math>X</math> and suppose <math>x \in X</math> is not an element of <math>U.</math> Then there exists a continuous linear functional <math>f</math> on <math>X</math> such that{{sfn|Narici|Beckenstein|2011|pp=177-220}}
<math>\sup |f(U)| \leq |f(x)|.</math>