Editing Hahn–Banach theorem (section)

==Generalizations==

'''General template'''

There are now many other versions of the Hahn–Banach theorem. The general template for the various versions of the Hahn–Banach theorem presented in this article is as follows:

:<math>p : X \to \R</math> is a [[sublinear function]] (possibly a [[seminorm]]) on a vector space <math>X,</math> <math>M</math> is a vector subspace of <math>X</math> (possibly closed), and <math>f</math> is a linear functional on <math>M</math> satisfying <math>|f| \leq p</math> on <math>M</math> (and possibly some other conditions). One then concludes that there exists a linear extension <math>F</math> of <math>f</math> to <math>X</math> such that <math>|F| \leq p</math> on <math>X</math> (possibly with additional properties).

{{Math theorem
| name = Theorem{{sfn|Narici|Beckenstein|2011|pp=177-220}}
| math_statement = If <math>D</math> is an [[Absorbing set|absorbing]] [[Absolutely convex set|disk]] in a real or complex vector space <math>X</math> and if <math>f</math> be a linear functional defined on a vector subspace <math>M</math> of <math>X</math> such that <math>|f| \leq 1</math> on <math>M \cap D,</math> then there exists a linear functional <math>F</math> on <math>X</math> extending <math>f</math> such that <math>|F| \leq 1</math> on <math>D.</math>
}}

===For seminorms===

{{Math theorem
| name = {{visible anchor|Hahn–Banach theorem for seminorms}}{{sfn|Wilansky|2013|pp=18-21}}{{sfn|Narici|Beckenstein|2011|pp=150}}
| math_statement = If <math>p : M \to \Reals</math> is a [[seminorm]] defined on a vector subspace <math>M</math> of <math>X,</math> and if <math>q : X \to \Reals</math> is a seminorm on <math>X</math> such that <math>p \leq q\big\vert_M,</math> then there exists a seminorm <math>P : X \to \Reals</math> on <math>X</math> such that <math>P\big\vert_M = p</math> on <math>M</math> and <math>P \leq q</math> on <math>X.</math>
}}

{{Math proof|title=Proof of the [[#Hahn–Banach theorem for seminorms|Hahn–Banach theorem for seminorms]]|drop=hidden|proof=
Let <math>S</math> be the convex hull of <math>\{m \in M : p(m) \leq 1\} \cup \{x \in X : q(x) \leq 1\}.</math> Because <math>S</math> is an [[Absorbing set|absorbing]] [[Absolutely convex set|disk]] in <math>X,</math> its [[Minkowski functional]] <math>P</math> is a seminorm. Then <math>p = P</math> on <math>M</math> and <math>P \leq q</math> on <math>X.</math>
}}

So for example, suppose that <math>f</math> is a [[bounded linear functional]] defined on a vector subspace <math>M</math> of a [[normed space]] <math>X,</math> so its the [[operator norm]] <math>\|f\|</math> is a non-negative real number. 
Then the linear functional's [[absolute value]] <math>p := |f|</math> is a seminorm on <math>M</math> and the map <math>q : X \to \Reals</math> defined by <math>q(x) = \|f\| \, \|x\|</math> is a seminorm on <math>X</math> that satisfies <math>p \leq q\big\vert_M</math> on <math>M.</math> 
The [[#Hahn–Banach theorem for seminorms|Hahn–Banach theorem for seminorms]] guarantees the existence of a seminorm <math>P : X \to \Reals</math> that is equal to <math>|f|</math> on <math>M</math> (since <math>P\big\vert_M = p = |f|</math>) and is bounded above by <math>P(x) \leq \|f\| \, \|x\|</math> everywhere on <math>X</math> (since <math>P \leq q</math>).

===Geometric separation===

{{Math theorem
| name = {{visible anchor|Hahn–Banach sandwich theorem}}{{sfn|Narici|Beckenstein|2011|pp=177-220}}
| math_statement = Let <math>p : X \to \R</math> be a sublinear function on a real vector space <math>X,</math> let <math>S \subseteq X</math> be any subset of <math>X,</math> and let <math>f : S \to \R</math> be {{em|any}} map. 
If there exist positive real numbers <math>a</math> and <math>b</math> such that 
<math display=block>0 \geq \inf_{s \in S} [p(s - a x - b y) - f(s) - a f(x) - b f(y)] \qquad \text{ for all } x, y \in S,</math>
then there exists a linear functional <math>F : X \to \R</math> on <math>X</math> such that <math>F \leq p</math> on <math>X</math> and <math>f \leq F \leq p</math> on <math>S.</math>
}}

===Maximal dominated linear extension===

{{Math theorem
| name = Theorem{{sfn|Narici|Beckenstein|2011|pp=177-220}}
| note = Andenaes, 1970
| math_statement = Let <math>p : X \to \R</math> be a sublinear function on a real vector space <math>X,</math> let <math>f : M \to \R</math> be a linear functional on a vector subspace <math>M</math> of <math>X</math> such that <math>f \leq p</math> on <math>M,</math> and let <math>S \subseteq X</math> be any subset of <math>X.</math> 
Then there exists a linear functional <math>F : X \to \R</math> on <math>X</math> that extends <math>f,</math> satisfies <math>F \leq p</math> on <math>X,</math> and is (pointwise) maximal on <math>S</math> in the following sense: if <math>\widehat{F} : X \to \R</math> is a linear functional on <math>X</math> that extends <math>f</math> and satisfies <math>\widehat{F} \leq p</math> on <math>X,</math> then <math>F \leq \widehat{F}</math> on <math>S</math> implies <math>F = \widehat{F}</math> on <math>S.</math>
}}

If <math>S = \{s\}</math> is a singleton set (where <math>s \in X</math> is some vector) and if <math>F : X \to \R</math> is such a maximal dominated linear extension of <math>f : M \to \R,</math> then <math>F(s) = \inf_{m \in M} [f(s) + p(s - m)].</math>{{sfn|Narici|Beckenstein|2011|pp=177-220}}

===Vector valued Hahn–Banach===
{{See also|Vector-valued Hahn–Banach theorems}}

{{Math theorem
| name = {{visible anchor|Vector–valued Hahn–Banach theorem}}{{sfn|Narici|Beckenstein|2011|pp=177-220}}
| math_statement = If <math>X</math> and <math>Y</math> are vector spaces over the same field and if <math>f : M \to Y</math> is a linear map defined on a vector subspace <math>M</math> of <math>X,</math> then there exists a linear map <math>F : X \to Y</math> that extends <math>f.</math>
}}

===Invariant Hahn–Banach===
{{See also|Vector-valued Hahn–Banach theorems}}

A set <math>\Gamma</math> of maps <math>X \to X</math> is {{em|{{visible anchor|commutative set of maps|text=commutative}}}} (with respect to [[function composition]] <math>\,\circ\,</math>) if <math>F \circ G = G \circ F</math> for all <math>F, G \in \Gamma.</math> 
Say that a function <math>f</math> defined on a subset <math>M</math> of <math>X</math> is {{em|{{visible anchor|invariant map|text=<math>\Gamma</math>-invariant}}}} if <math>L(M) \subseteq M</math> and <math>f \circ L = f</math> on <math>M</math> for every <math>L \in \Gamma.</math>

{{Math theorem
| name = {{visible anchor|An invariant Hahn–Banach theorem}}{{sfn|Rudin|1991|p=141}}
| math_statement = 
Suppose <math>\Gamma</math> is a [[#commutative set of maps|commutative set]] of continuous linear maps from a [[normed space]] <math>X</math> into itself and let <math>f</math> be a continuous linear functional defined some vector subspace <math>M</math> of <math>X</math> that is [[#invariant map|<math>\Gamma</math>-invariant]], which means that <math>L(M) \subseteq M</math> and <math>f \circ L = f</math> on <math>M</math> for every <math>L \in \Gamma.</math> 
Then <math>f</math> has a continuous linear extension <math>F</math> to all of <math>X</math> that has the same [[operator norm]] <math>\|f\| = \|F\|</math> and is also <math>\Gamma</math>-invariant, meaning that <math>F \circ L = F</math> on <math>X</math> for every <math>L \in \Gamma.</math>
}}

This theorem may be summarized:

:Every [[#invariant map|<math>\Gamma</math>-invariant]] continuous linear functional defined on a vector subspace of a normed space <math>X</math> has a <math>\Gamma</math>-invariant Hahn–Banach extension to all of <math>X.</math>{{sfn|Rudin|1991|p=141}}

===For nonlinear functions===
The following theorem of Mazur–Orlicz (1953) is equivalent to the Hahn–Banach theorem.

{{Math theorem
| name = {{visible anchor|Mazur–Orlicz theorem}}{{sfn|Narici|Beckenstein|2011|pp=177–220}}
| math_statement = Let <math>p : X \to \R</math> be a [[sublinear function]] on a real or complex vector space <math>X,</math> let <math>T</math> be any set, and let <math>R : T \to \R</math> and <math>v : T \to X</math> be any maps. The following statements are equivalent:
# there exists a real-valued linear functional <math>F</math> on <math>X</math> such that <math>F \leq p</math> on <math>X</math> and <math>R \leq F \circ v</math> on <math>T</math>;
# for any finite sequence <math>s_1, \ldots, s_n</math> of <math>n > 0</math> non-negative real numbers, and any sequence <math>t_1, \ldots, t_n \in T</math> of elements of <math>T,</math> <math display=block>\sum_{i=1}^n s_i R\left(t_i\right) \leq p\left(\sum_{i=1}^n s_i v\left(t_i\right)\right).</math>
}}

The following theorem characterizes when {{em|any}} scalar function on <math>X</math> (not necessarily linear) has a continuous linear extension to all of <math>X.</math>

{{Math theorem
| name = Theorem
| note = {{visible anchor|The extension principle}}{{sfn|Edwards|1995|pp=124-125}}
| math_statement = Let <math>f</math> a scalar-valued function on a subset <math>S</math> of a [[topological vector space]] <math>X.</math>  
Then there exists a continuous linear functional <math>F</math> on <math>X</math> extending <math>f</math> if and only if there exists a continuous seminorm <math>p</math> on <math>X</math> such that 
<math display=block>\left|\sum_{i=1}^n a_i f(s_i)\right| \leq p\left(\sum_{i=1}^n a_is_i\right)</math>
for all positive integers <math>n</math> and all finite sequences <math>a_1, \ldots, a_n</math> of scalars and elements <math>s_1, \ldots, s_n</math> of <math>S.</math>
}}