Editing Hahn–Banach theorem

{{short description|Theorem on extension of bounded linear functionals}}
In [[functional analysis]], the '''Hahn–Banach theorem''' is a central result that allows the extension of [[Bounded operator|bounded linear functionals]] defined on a [[vector subspace]] of some [[vector space]] to the whole space.   The theorem also shows that there are sufficient [[Continuous function (topology)|continuous]] linear functionals defined on every [[normed vector space]] in order to study the [[dual space]]. Another version of the Hahn–Banach theorem is known as the '''Hahn–Banach separation theorem''' or the [[hyperplane separation theorem]], and has numerous uses in [[convex geometry]].

==History==

The theorem is named for the mathematicians [[Hans Hahn (mathematician)|Hans Hahn]] and [[Stefan Banach]], who proved it independently in the late 1920s. 
The special case of the theorem for the space <math>C[a, b]</math> of continuous functions on an interval was proved earlier (in 1912) by [[Eduard Helly]],<ref>{{MacTutor Biography|id=Helly}}</ref> and a more general extension theorem, the [[M. Riesz extension theorem]], from which the Hahn–Banach theorem can be derived, was proved in 1923 by [[Marcel Riesz]].<ref>See [[M. Riesz extension theorem]]. 
According to {{cite journal|mr=0256837|last=Gårding|first=L.|author-link=Lars Gårding|title=Marcel Riesz in memoriam|journal=[[Acta Math.]]|volume=124|year=1970|issue=1|pages=I–XI |doi=10.1007/bf02394565|doi-access=free}}, the argument was known to Riesz already in 1918.</ref>

The first Hahn–Banach theorem was proved by [[Eduard Helly]] in 1912 who showed that certain linear functionals defined on a subspace of a certain type of normed space (<math>\Complex^{\N}</math>) had an extension of the same norm. Helly did this through the technique of first proving that a one-dimensional [[Extension of a function|extension]] exists (where the linear functional has its domain extended by one dimension) and then using [[Mathematical induction|induction]].  In 1927, Hahn defined general [[Banach space]]s and used Helly's technique to prove a norm-preserving version of Hahn–Banach theorem for Banach spaces (where a bounded linear functional on a subspace has a bounded linear extension of the same norm to the whole space).  In 1929, Banach, who was unaware of Hahn's result, generalized it by replacing the norm-preserving version with the dominated extension version that uses [[sublinear function]]s.  Whereas Helly's proof used mathematical induction, Hahn and Banach both used [[transfinite induction]].{{sfn|Narici|Beckenstein|2011|pp=177-220}}

The Hahn–Banach theorem arose from attempts to solve infinite systems of linear equations. This is needed to solve problems such as the [[moment problem]], whereby given all the potential [[Moment (mathematics)|moments of a function]] one must determine if a function having these moments exists, and, if so, find it in terms of those moments. Another such problem is the [[Fourier sine and cosine series|Fourier cosine series]] problem, whereby given all the potential Fourier cosine coefficients one must determine if a function having those coefficients exists, and, again, find it if so.

Riesz and Helly solved the problem for certain classes of spaces (such as [[Lp space|<math>L^p([0, 1])</math>]] and [[Continuous functions on a compact Hausdorff space|<math>C([a, b])</math>]]) where they discovered that the existence of a solution was equivalent to the existence and continuity of certain linear functionals.  In effect, they needed to solve the following problem:{{sfn|Narici|Beckenstein|2011|pp=177-220}} 

:('''{{visible anchor|The vector problem}}''') Given a collection <math>\left(f_i\right)_{i \in I}</math> of bounded linear functionals on a [[normed space]] <math>X</math> and a collection of scalars <math>\left(c_i\right)_{i \in I},</math> determine if there is an <math>x \in X</math> such that <math>f_i(x) = c_i</math> for all <math>i \in I.</math>

If <math>X</math> happens to be a [[reflexive space]] then to solve the vector problem, it suffices to solve the following dual problem:{{sfn|Narici|Beckenstein|2011|pp=177-220}}

:('''The functional problem''') Given a collection <math>\left(x_i\right)_{i \in I}</math> of vectors in a normed space <math>X</math> and a collection of scalars <math>\left(c_i\right)_{i \in I},</math> determine if there is a bounded linear functional <math>f</math> on <math>X</math> such that <math>f\left(x_i\right) = c_i</math> for all <math>i \in I.</math>

Riesz went on to define [[Lp space|<math>L^p([0, 1])</math> space]] (<math>1 < p < \infty</math>) in 1910 and the <math>\ell^p</math> spaces in 1913.  While investigating these spaces he proved a special case of the Hahn–Banach theorem.  Helly also proved a special case of the Hahn–Banach theorem in 1912.  In 1910, Riesz solved the functional problem for some specific spaces and in 1912, Helly solved it for a more general class of spaces. It wasn't until 1932 that Banach, in one of the first important applications of the Hahn–Banach theorem, solved the general functional problem.  The following theorem states the general functional problem and characterizes its solution.{{sfn|Narici|Beckenstein|2011|pp=177-220}}

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=177-220}}|note=The functional problem|math_statement=
Let <math>\left(x_i\right)_{i \in I}</math> be vectors in a [[real number|real]] or [[complex number|complex]] normed space <math>X</math> and let <math>\left(c_i\right)_{i \in I}</math> be scalars also [[Index set|indexed by]] <math>I \neq \varnothing.</math>

There exists a continuous linear functional <math>f</math> on <math>X</math> such that <math>f\left(x_i\right) = c_i</math> for all <math>i \in I</math> if and only if there exists a <math>K > 0</math> such that for any choice of scalars <math>\left(s_i\right)_{i \in I}</math> where all but finitely many <math>s_i</math> are <math>0,</math> the following holds:
<math display=block>\left|\sum_{i \in I} s_i c_i\right| \leq K \left\|\sum_{i \in I} s_i x_i\right\|.</math>
}}

The Hahn–Banach theorem can be deduced from the above theorem.{{sfn|Narici|Beckenstein|2011|pp=177-220}} If <math>X</math> is [[Reflexive space|reflexive]] then this theorem solves the vector problem.

==Hahn–Banach theorem==

A real-valued function <math>f : M \to \R</math> defined on a subset <math>M</math> of <math>X</math> is said to be {{em|{{visible anchor|dominated real functional|text=dominated (above) by}}}} a function <math>p : X \to \R</math> if <math>f(m) \leq p(m)</math> for every <math>m \in M.</math> 
For this reason, the following version of the Hahn–Banach theorem is called {{em|the dominated [[Extension of a function|extension]] theorem}}. 

{{Math theorem
| name = {{visible anchor|Hahn–Banach dominated extension theorem}} (for real linear functionals){{sfn|Rudin|1991|pp=56-62}}<ref>{{harvnb|Rudin|1991}}, Th. 3.2</ref>{{sfn|Narici|Beckenstein|2011|pp=177-183}}
| math_statement =
If <math>p : X \to \R</math> is a [[sublinear function]] (such as a [[Norm (mathematics)|norm]] or [[seminorm]] for example) defined on a real vector space <math>X</math> then any [[linear functional]] defined on a vector subspace of <math>X</math> that is [[#dominated real functional|dominated above]] by <math>p</math> has at least one [[Linear extension (linear algebra)|linear extension]] to all of <math>X</math> that is also dominated above by <math>p.</math> 

Explicitly, if <math>p : X \to \R</math> is a [[sublinear function]], which by definition means that it satisfies 
<math display=block>p(x + y) \leq p(x) + p(y) \quad \text{ and } \quad p(t x) = t p(x) \qquad \text{ for all } \; x, y \in X \; \text{ and all real } \; t \geq 0,</math> 
and if <math>f : M \to \R</math> is a linear functional defined on a vector subspace <math>M</math> of <math>X</math> such that 
<math display=block>f(m) \leq p(m) \quad \text{ for all } m \in M</math> 
then there exists a linear functional <math>F : X \to \R</math> such that
<math display=block>F(m) = f(m) \quad \text{ for all } m \in M,</math>
<math display=block>F(x) \leq p(x) \quad ~\;\, \text{ for all } x \in X.</math> 
Moreover, if <math>p</math> is a [[seminorm]] then <math>|F(x)| \leq p(x)</math> necessarily holds for all <math>x \in X.</math> 
}}

The theorem remains true if the requirements on <math>p</math> are relaxed to require only that <math>p</math> be a [[convex function]]:{{Sfn|Schechter|1996|pp=318-319}}{{Sfn|Reed|Simon|1980|p=}}
<math display=block>p(t x + (1 - t) y) \leq t p(x) + (1 - t) p(y) \qquad \text{ for all } 0 < t < 1 \text{ and } x, y \in X.</math> 
<!-- Substituting <math>\tfrac{t}{1-t} y</math> in for <math>y</math> gives <math>p(t x + t y) \leq t p(x) + (1 - t) p\left(\tfrac{t}{1-t} y\right).</math> Substituting <math>\tfrac{1}{t} x</math> in for <math>x</math> and <math>\tfrac{1}{1-t} y</math> in for <math>y</math> gives <math>p(x + y) \leq \tfrac{p((1/t) x)}{1/t} + \tfrac{p(1/(1-t) y)}{1/(1-t)}.</math> --> 
A function <math>p : X \to \R</math> is convex and satisfies <math>p(0) \leq 0</math> if and only if <math>p(a x + b y) \leq a p(x) + b p(y)</math> for all vectors <math>x, y \in X</math> and all non-negative real <math>a, b \geq 0</math> such that <math>a + b \leq 1.</math> Every [[sublinear function]] is a convex function. 
On the other hand, if <math>p : X \to \R</math> is convex with <math>p(0) \geq 0,</math> then the function defined by <math>p_0(x) \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \inf_{t > 0} \frac{p(tx)}{t}</math> is [[Homogeneous_function#Positive_homogeneity|positively homogeneous]] 
(because for all <math>x</math> and <math>r>0</math> one has <math>p_0(rx)=\inf_{t > 0} \frac{p(trx)}{t} =r\inf_{t > 0} \frac{p(trx)}{tr} = r\inf_{\tau > 0} \frac{p(\tau x)}{\tau}=rp_0(x)</math>), hence, being convex, [[Sublinear_function#Examples_and_sufficient_conditions|it is sublinear]]. It is also bounded above by <math>p_0 \leq p,</math> and satisfies <math>F \leq p_0</math> for every linear functional <math>F \leq p.</math> So the extension of the Hahn–Banach theorem to convex functionals does not have a much larger content than the classical one stated for sublinear functionals.

If <math>F : X \to \R</math> is linear then <math>F \leq p</math> if and only if{{sfn|Rudin|1991|pp=56-62}} <math display=block>-p(-x) \leq F(x) \leq p(x) \quad \text{ for all } x \in X,</math> 
which is the (equivalent) conclusion that some authors{{sfn|Rudin|1991|pp=56-62}} write instead of <math>F \leq p.</math>
It follows that if <math>p : X \to \R</math> is also {{em|symmetric}}, meaning that <math>p(-x) = p(x)</math> holds for all <math>x \in X,</math> then <math>F \leq p</math> if and only <math>|F| \leq p.</math> 
Every [[Norm (mathematics)|norm]] is a [[seminorm]] and both are symmetric [[Balanced function|balanced]] sublinear functions. A sublinear function is a seminorm if and only if it is a [[balanced function]]. On a real vector space (although not on a complex vector space), a sublinear function is a seminorm if and only if it is symmetric. The [[identity function]] <math>\R \to \R</math> on <math>X := \R</math> is an example of a sublinear function that is not a seminorm. 

===For complex or real vector spaces===

The dominated extension theorem for real linear functionals implies the following alternative statement of the Hahn–Banach theorem that can be applied to linear functionals on real or complex vector spaces. 

{{Math theorem
| name = {{visible anchor|Hahn–Banach theorem for real or complex vector spaces|text=Hahn–Banach theorem}}{{sfn|Narici|Beckenstein|2011|pp=177-220}}<ref>{{harvnb|Rudin|1991}}, Th. 3.2</ref>
| math_statement = Suppose <math>p : X \to \R</math> a [[seminorm]] on a vector space <math>X</math> over the field <math>\mathbf{K},</math> which is either <math>\R</math> or <math>\Complex.</math> 
If <math>f : M \to \mathbf{K}</math> is a linear functional on a vector subspace <math>M</math> such that 
<math display=block>|f(m)| \leq p(m) \quad \text{ for all } m \in M,</math>
then there exists a linear functional <math>F : X \to \mathbf{K}</math> such that
<math display=block>F(m) = f(m) \quad \; \text{ for all } m \in M,</math>
<math display=block>|F(x)| \leq p(x) \quad \;\, \text{ for all } x \in X.</math>
}}

The theorem remains true if the requirements on <math>p</math> are relaxed to require only that for all <math>x, y \in X</math> and all scalars <math>a</math> and <math>b</math> satisfying <math>|a| + |b| \leq 1,</math>{{Sfn|Reed|Simon|1980|p=}}
<math display=block>p(a x + b y) \leq |a| p(x) + |b| p(y).</math>
This condition holds if and only if <math>p</math> is a [[Convex function|convex]] and [[balanced function]] satisfying <math>p(0) \leq 0,</math> or equivalently, if and only if it is convex, satisfies <math>p(0) \leq 0,</math> and <math>p(u x) \leq p(x)</math> for all <math>x \in X</math> and all [[unit length]] scalars <math>u.</math> 

A complex-valued functional <math>F</math> is said to be {{em|{{visible anchor|dominated complex functional|text=dominated by <math>p</math>}}}} if <math>|F(x)| \leq p(x)</math> for all <math>x</math> in the domain of <math>F.</math> 
With this terminology, the above statements of the Hahn–Banach theorem can be restated more succinctly:

:'''Hahn–Banach dominated extension theorem''': If <math>p : X \to \R</math> is a [[seminorm]] defined on a real or complex vector space <math>X,</math> then every [[#dominated complex functional|dominated]] linear functional defined on a vector subspace of <math>X</math> has a dominated linear extension to all of <math>X.</math> In the case where <math>X</math> is a real vector space and <math>p : X \to \R</math> is merely a [[Convex function|convex]] or [[sublinear function]], this conclusion will remain true if both instances of "[[#dominated complex functional|dominated]]" (meaning <math>|F| \leq p</math>) are weakened to instead mean "[[#dominated real functional|dominated {{em|above}}]]" (meaning <math>F \leq p</math>).{{Sfn|Schechter|1996|pp=318-319}}{{Sfn|Reed|Simon|1980|p=}} 

'''Proof'''

The following observations allow the [[#Hahn–Banach dominated extension theorem|Hahn–Banach theorem for real vector spaces]] to be applied to (complex-valued) linear functionals on complex vector spaces.

Every linear functional <math>F : X \to \Complex</math> on a complex vector space is [[Linear form#Real and imaginary parts of a linear functional|completely determined]] by its [[real part]] <math>\; \operatorname{Re} F : X \to \R \;</math> through the formula{{sfn|Narici|Beckenstein|2011|pp=177-183}}<ref group=proof>If <math>z = a + i b \in \Complex</math> has real part <math>\operatorname{Re} z = a</math> then <math>- \operatorname{Re} (i z) = b,</math> which proves that <math>z = \operatorname{Re} z - i \operatorname{Re} (i z).</math> Substituting <math>F(x)</math> in for <math>z</math> and using <math>i F(x) = F(i x)</math> gives <math>F(x) = \operatorname{Re} F(x) - i \operatorname{Re} F(i x).</math> <math>\blacksquare</math></ref> 
<math display=block>F(x) \;=\; \operatorname{Re} F(x) - i \operatorname{Re} F(i x) \qquad \text{ for all } x \in X</math>
and moreover, if <math>\|\cdot\|</math> is a [[Norm (mathematics)|norm]] on <math>X</math> then their [[dual norm]]s are equal: <math>\|F\| = \|\operatorname{Re} F\|.</math>{{sfn|Narici|Beckenstein|2011|pp=126-128}} 
In particular, a linear functional on <math>X</math> extends another one defined on <math>M \subseteq X</math> if and only if their real parts are equal on <math>M</math> (in other words, a linear functional <math>F</math> extends <math>f</math> if and only if <math>\operatorname{Re} F</math> extends <math>\operatorname{Re} f</math>). 
The real part of a linear functional on <math>X</math> is always a {{visible anchor|real-linear functional}} (meaning that it is linear when <math>X</math> is considered as a real vector space) and if <math>R : X \to \R</math> is a real-linear functional on a complex vector space then <math>x \mapsto R(x) - i R(i x)</math> defines the unique linear functional on <math>X</math> whose real part is <math>R.</math> 

If <math>F</math> is a linear functional on a (complex or real) vector space <math>X</math> and if <math>p : X \to \R</math> is a seminorm then{{sfn|Narici|Beckenstein|2011|pp=177-183}}<ref group=proof>Let <math>F</math> be any [[Homogeneous function#Homogeneity|homogeneous]] scalar-valued map on <math>X</math> (such as a linear functional) and let <math>p : X \to \R</math> be any map that satisfies <math>p(u x) = p(x)</math> for all <math>x</math> and [[unit length]] scalars <math>u</math> (such as a seminorm). If <math>|F| \leq p</math> then <math>\operatorname{Re} F \leq |\operatorname{Re} F| \leq |F| \leq p.</math> For the converse, assume <math>\operatorname{Re} F \leq p</math> and fix <math>x \in X.</math> Let <math>r = |F(x)|</math> and pick any <math>\theta \in \R</math> such that <math>F(x) = r e^{i \theta};</math> it remains to show <math>r \leq p(x).</math> Homogeneity of <math>F</math> implies <math>F\left(e^{-i \theta} x\right) = r</math> is real so that <math>\operatorname{Re} F\left(e^{-i \theta} x\right) = F\left(e^{-i \theta} x\right).</math> By assumption, <math>\operatorname{Re} F \leq p</math> and <math>p\left(e^{-i \theta} x\right) = p(x),</math> so that <math>r = \operatorname{Re} F\left(e^{-i \theta} x\right) \leq p\left(e^{-i \theta} x\right) = p(x),</math> as desired. <math>\blacksquare</math></ref> 
<math display=block>|F| \,\leq\, p \quad \text{ if and only if } \quad \operatorname{Re} F \,\leq\, p.</math>
Stated in simpler language, a linear functional is [[#dominated complex functional|dominated]] by a seminorm <math>p</math> if and only if its [[#dominated real functional|real part is dominated above]] by <math>p.</math> 

{{Math proof|title=Proof of [[#Hahn–Banach theorem for real or complex vector spaces|Hahn–Banach for complex vector spaces]] by reduction to real vector spaces{{sfn|Narici|Beckenstein|2011|pp=177-220}}|drop=hidden|proof=
Suppose <math>p : X \to \R</math> is a seminorm on a complex vector space <math>X</math> and let <math>f : M \to \Complex</math> be a linear functional defined on a vector subspace <math>M</math> of <math>X</math> that satisfies <math>|f| \leq p</math> on <math>M.</math> 
Consider <math>X</math> as a real vector space and apply the [[#Hahn–Banach dominated extension theorem|Hahn–Banach theorem for real vector spaces]] to the [[#real-linear functional|real-linear functional]] <math>\; \operatorname{Re} f : M \to \R \;</math> to obtain a real-linear extension <math>R : X \to \R</math> that is also dominated above by <math>p,</math> so that it satisfies <math>R \leq p</math> on <math>X</math> and <math>R = \operatorname{Re} f</math> on <math>M.</math> 
The map <math>F : X \to \Complex</math> defined by <math>F(x) \;=\; R(x) - i R(i x)</math> is a linear functional on <math>X</math> that extends <math>f</math> (because their real parts agree on <math>M</math>) and satisfies <math>|F| \leq p</math> on <math>X</math> (because <math>\operatorname{Re} F \leq p</math> and <math>p</math> is a seminorm). 
<math>\blacksquare</math>
}}

The proof above shows that when <math>p</math> is a seminorm then there is a one-to-one correspondence between dominated linear extensions of <math>f : M \to \Complex</math> and dominated real-linear extensions of <math>\operatorname{Re} f : M \to \R;</math> the proof even gives a formula for explicitly constructing a linear extension of <math>f</math> from any given real-linear extension of its real part.

'''Continuity'''

A linear functional <math>F</math> on a [[topological vector space]]  is [[Continuous linear functional|continuous]] if and only if this is true of its real part <math>\operatorname{Re} F;</math> if the domain is a normed space then <math>\|F\| = \|\operatorname{Re} F\|</math> (where one side is infinite if and only if the other side is infinite).{{sfn|Narici|Beckenstein|2011|pp=126-128}} 
Assume <math>X</math> is a [[topological vector space]] and <math>p : X \to \R</math> is [[sublinear function]]. 
If <math>p</math> is a [[Continuity (topology)|continuous]] sublinear function that dominates a linear functional <math>F</math> then <math>F</math> is necessarily continuous.{{sfn|Narici|Beckenstein|2011|pp=177-183}} Moreover, a linear functional <math>F</math> is continuous if and only if its [[absolute value]] <math>|F|</math> (which is a [[seminorm]] that dominates <math>F</math>) is continuous.{{sfn|Narici|Beckenstein|2011|pp=177-183}} In particular, a linear functional is continuous if and only if it is dominated by some continuous sublinear function.

===Proof===

The [[#Hahn–Banach dominated extension theorem|Hahn–Banach theorem for real vector spaces]] ultimately follows from Helly's initial result for the special case where the linear functional is extended from <math>M</math> to a larger vector space in which <math>M</math> has [[codimension]] <math>1.</math>{{sfn|Narici|Beckenstein|2011|pp=177-220}}

{{Math theorem
| name = Lemma{{sfn|Narici|Beckenstein|2011|pp=177-183}}
| note = {{visible anchor|One–dimensional dominated extension theorem}}
| 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> a [[linear functional]] on a [[Proper subset|proper]] [[vector subspace]] <math>M \subsetneq X</math> such that <math>f \leq p</math> on <math>M</math> (meaning <math>f(m) \leq p(m)</math> for all <math>m \in M</math>), and let <math>x \in X</math> be a vector {{em|not}} in <math>M</math> (so <math>M \oplus \R x = \operatorname{span} \{M, x\}</math>). 
There exists a linear extension <math>F : M \oplus \R x \to \R</math> of <math>f</math> such that <math>F \leq p</math> on <math>M \oplus \R x.</math>
}}

{{Math proof|title=Proof{{sfn|Narici|Beckenstein|2011|pp=177-183}}|drop=hidden|proof=
Given any real number <math>b,</math> the map <math>F_b : M \oplus \R x \to \R</math> defined by <math>F_b(m + r x) = f(m) + r b</math> is always a linear extension of <math>f</math> to <math>M \oplus \R x</math><ref group=note>This definition means, for instance, that <math>F_b(x) = F_b(0 + 1 x) = f(0) + 1 b = b</math> and if <math>m \in M</math> then <math>F_b(m) = F_b(m + 0 x) = f(m) + 0 b = f(m).</math> In fact, if <math>G : M \oplus \R x \to \R</math> is any linear extension of <math>f</math> to <math>M \oplus \R x</math> then <math>G = F_b</math> for <math>b := G(x).</math> In other words, every linear extension of <math>f</math> to <math>M \oplus \R x</math> is of the form <math>F_b</math> for some (unique) <math>b.</math></ref> but it might not satisfy <math>F_b \leq p.</math> 
It will be shown that <math>b</math> can always be chosen so as to guarantee that <math>F_b \leq p,</math> which will complete the proof. 

If <math>m, n \in M</math> then 
<math display=block>f(m) - f(n) = f(m - n) \leq p(m - n) = p(m + x - x - n) \leq p(m + x) + p(- x - n)</math>
which implies
<math display=block>-p(-n - x) - f(n) ~\leq~ p(m + x) - f(m).</math>
<!--where importantly, the left hand side is independent of <math>m</math> and the right hand side is independent of <math>n.</math>--> 
So define
<math display=block>a = \sup_{n \in M}[-p(-n - x) - f(n)] \qquad \text{ and } \qquad c = \inf_{m \in M} [p(m + x) - f(m)]</math>
where <math>a \leq c</math> are real numbers. 
To guarantee <math>F_b \leq p,</math> it suffices that <math>a \leq b \leq c</math> (in fact, this is also necessary<ref group=note>Explicitly, for any real number <math>b \in \R,</math> <math>F_b \leq p</math> on <math>M \oplus \R x</math> if and only if <math>a \leq b \leq c.</math> Combined with the fact that <math>F_b(x) = b,</math> it follows that the dominated linear extension of <math>f</math> to <math>M \oplus \R x</math> is unique if and only if <math>a = c,</math> in which case this scalar will be the extension's values at <math>x.</math> Since every linear extension of <math>f</math> to <math>M \oplus \R x</math> is of the form <math>F_b</math> for some <math>b,</math> the bounds <math>a \leq b = F_b(x) \leq c</math> thus also limit the range of possible values (at <math>x</math>) that can be taken by any of <math>f</math>'s dominated linear extensions. Specifically, if <math>F : X \to \R</math> is any linear extension of <math>f</math> satisfying <math>F \leq p</math> then for every <math>x \in X \setminus M,</math> <math>\sup_{m \in M}[-p(-m - x) - f(m)] ~\leq~ F(x) ~\leq~ \inf_{m \in M} [p(m + x) - f(m)].</math></ref>) because then <math>b</math> satisfies "the decisive inequality"{{sfn|Narici|Beckenstein|2011|pp=177-183}}
<math display=block>-p(-n - x) - f(n) ~\leq~ b ~\leq~ p(m + x) - f(m) \qquad \text{ for all }\;  m, n \in M.</math>

To see that <math>f(m) + r b \leq p(m + r x)</math> follows,<ref group=note name="GeometricIllustration" /> assume <math>r \neq 0</math> and substitute <math>\tfrac{1}{r} m</math> in for both <math>m</math> and <math>n</math> to obtain
<math display=block>-p\left(- \tfrac{1}{r} m - x\right) - \tfrac{1}{r} f\left(m\right) ~\leq~ b ~\leq~ p\left(\tfrac{1}{r} m + x\right) - \tfrac{1}{r} f\left(m\right).</math> 
If <math>r > 0</math> (respectively, if <math>r < 0</math>) then the right (respectively, the left) hand side equals <math>\tfrac{1}{r} \left[p(m + r x) - f(m)\right]</math> so that multiplying by <math>r</math> gives <math>r b \leq p(m + r x) - f(m).</math> 
<!--<math display=block>-p\left(- \tfrac{1}{r} m - x\right) - f\left(\tfrac{1}{r} m\right) = -p\left(\left(- \tfrac{1}{r}\right) \left(m + rx\right)\right) - \tfrac{1}{r} f(m) = \tfrac{1}{r} \left[p(m + r x) - f(m)\right].</math> -->
<math>\blacksquare</math>
}}

This lemma remains true if <math>p : X \to \R</math> is merely a [[convex function]] instead of a sublinear function.{{Sfn|Schechter|1996|pp=318-319}}{{Sfn|Reed|Simon|1980|p=}} 

{{collapse top|title=Proof|left=true}}
Assume that <math>p</math> is convex, which means that <math>p(t y + (1 - t) z) \leq t p(y) + (1 - t) p(z)</math> for all <math>0 \leq t \leq 1</math> and <math>y, z \in X.</math> Let <math>M,</math> <math>f : M \to \R,</math> and <math>x \in X \setminus M</math> be as in [[Hahn–Banach theorem#One–dimensional dominated extension theorem|the lemma's statement]]. Given any <math>m, n \in M</math> and any positive real <math>r, s > 0,</math> the positive real numbers <math>t := \tfrac{s}{r + s}</math> and <math>\tfrac{r}{r + s} = 1 - t</math> sum to <math>1</math> so that the convexity of <math>p</math> on <math>X</math> guarantees
<math display=block>\begin{alignat}{9}
p\left(\tfrac{s}{r + s} m + \tfrac{r}{r + s} n\right) 
~&=~ p\big(\tfrac{s}{r + s} (m - r x) &&+ \tfrac{r}{r + s} (n + s x)\big) &&  \\
&\leq~ \tfrac{s}{r + s} \; p(m - r x) &&+ \tfrac{r}{r + s} \; p(n + s x)  && \\
\end{alignat}</math>
and hence
<math display=block>\begin{alignat}{9}
s f(m) + r f(n) 
~&=~ (r + s) \; f\left(\tfrac{s}{r + s} m + \tfrac{r}{r + s} n\right)   && \qquad \text{ by linearity of } f \\
&\leq~ (r + s) \; p\left(\tfrac{s}{r + s} m + \tfrac{r}{r + s} n\right) && \qquad f \leq p \text{ on } M \\
&\leq~ s p(m - r x) + r p(n + s x) \\
\end{alignat}</math>
thus proving that <math>- s p(m - r x) + s f(m) ~\leq~ r p(n + s x) - r f(n),</math> which after multiplying both sides by <math>\tfrac{1}{rs}</math> becomes 
<math display=block>\tfrac{1}{r} [- p(m - r x) + f(m)] ~\leq~ \tfrac{1}{s} [p(n + s x) - f(n)].</math> 
This implies that the values defined by 
<math display=block>a = \sup_{\stackrel{m \in M}{r > 0}} \tfrac{1}{r} [- p(m - r x) + f(m)] \qquad \text{ and } \qquad c = \inf_{\stackrel{n \in M}{s > 0}} \tfrac{1}{s} [p(n + s x) - f(n)]</math> 
are real numbers that satisfy <math>a \leq c.</math> As in the above proof of the [[Hahn–Banach theorem#One–dimensional dominated extension theorem|one–dimensional dominated extension theorem]] above, for any real <math>b \in \R</math> define <math>F_b : M \oplus \R x \to \R</math> by <math>F_b(m + r x) = f(m) + r b.</math> 
It can be verified that if <math>a \leq b \leq c</math> then <math>F_b \leq p</math> where <math>r b \leq p(m + r x) - f(m)</math> follows from <math>b \leq c</math> when <math>r > 0</math> (respectively, follows from <math>a \leq b</math> when <math>r < 0</math>).
<math>\blacksquare</math>
{{collapse bottom}}

The [[#One–dimensional dominated extension theorem|lemma above]] is the key step in deducing the dominated extension theorem from [[Zorn's lemma]]. 

{{Math proof|title=Proof of dominated extension theorem using [[Zorn's lemma]]|drop=hidden|proof=
The set of all possible dominated linear extensions of <math>f</math> are partially ordered by extension of each other, so there is a maximal extension <math>F.</math>  By the codimension-1 result, if <math>F</math> is not defined on all of <math>X,</math> then it can be further extended.  Thus <math>F</math> must be defined everywhere, as claimed. 
<math>\blacksquare</math>
}}

When <math>M</math> has countable codimension, then using induction and the lemma completes the proof of the Hahn–Banach theorem. The standard proof of the general case uses [[Zorn's lemma]] although the strictly weaker [[ultrafilter lemma]]{{sfn|Luxemburg|1962|p=}} (which is equivalent to the [[compactness theorem]] and to the [[Boolean prime ideal theorem]]) may be used instead. Hahn–Banach can also be proved using [[Tychonoff's theorem]] for [[Compact space|compact]] [[Hausdorff space]]s{{sfn|Łoś|Ryll-Nardzewski|1951|pp=233–237}} (which is also equivalent to the ultrafilter lemma)

The [[Mizar system|Mizar project]] has completely formalized and automatically checked the proof of the Hahn–Banach theorem in the HAHNBAN file.<ref>[http://mizar.uwb.edu.pl/JFM/Vol5/hahnban.html HAHNBAN file]</ref>

==Continuous extension theorem==

The Hahn–Banach theorem can be used to guarantee the existence of [[Continuous linear operator|continuous linear extensions]] of [[continuous linear functional]]s.

{{Math theorem
|name={{visible anchor|Hahn–Banach continuous extension theorem}}{{sfn|Narici|Beckenstein|2011|pp=182,498}}
|math_statement=
Every continuous linear functional <math>f</math> defined on a vector subspace <math>M</math> of a (real or complex) [[Locally convex topological vector space|locally convex]] [[topological vector space]] <math>X</math> has a continuous linear extension <math>F</math> to all of <math>X.</math> If in addition <math>X</math> is a [[normed space]], then this extension can be chosen so that its [[dual norm]] is equal to that of <math>f.</math>
}}

In [[Category theory|category-theoretic]] terms, the underlying field of the vector space is an [[injective object]] in the category of locally convex vector spaces.

On a [[normed space|normed]] (or [[Seminormed space|seminormed]]) space, a linear extension <math>F</math> of a [[bounded linear functional]] <math>f</math> is said to be {{em|{{visible anchor|norm-preserving linear extension|text=norm-preserving}}}} if it has the same [[dual norm]] as the original functional: <math>\|F\| = \|f\|.</math>
Because of this terminology, the second part of [[#Hahn–Banach continuous extension theorem|the above theorem]] is sometimes referred to as the "[[#Norm-preserving Hahn–Banach continuous extension theorem|norm-preserving]]" version of the Hahn–Banach theorem.{{sfn|Narici|Beckenstein|2011|p=184}} Explicitly:

{{Math theorem
|name={{visible anchor|Norm-preserving Hahn–Banach continuous extension theorem}}{{sfn|Narici|Beckenstein|2011|p=184}}
|math_statement=Every continuous linear functional <math>f</math> defined on a vector subspace <math>M</math> of a (real or complex) normed space <math>X</math> has a continuous linear extension <math>F</math> to all of <math>X</math> that satisfies <math>\|f\| = \|F\|.</math>
}}

===Proof of the continuous extension theorem===

The following observations allow the [[#Hahn–Banach continuous extension theorem|continuous extension theorem]] to be deduced from the [[#Hahn–Banach theorem for real or complex vector spaces|Hahn–Banach theorem]].{{sfn|Narici|Beckenstein|2011|p=182}} 

The absolute value of a linear functional is always a seminorm. A linear functional <math>F</math> on a [[topological vector space]] <math>X</math> is continuous if and only if its absolute value <math>|F|</math> is continuous, which happens if and only if there exists a continuous seminorm <math>p</math> on <math>X</math> such that <math>|F| \leq p</math> on the domain of <math>F.</math>{{sfn|Narici|Beckenstein|2011|p=126}} 
If <math>X</math> is a locally convex space then this statement remains true when the linear functional <math>F</math> is defined on a {{em|proper}} vector subspace of <math>X.</math> 
<!-- Suppose <math>X</math> is a locally convex space whose topology is generated by a family <math>\mathcal{P}</math> of (necessarily continuous) seminorms (if <math>X</math> is a normed space then <math>\mathcal{P}</math> may be <math>\{\|\cdot\|\},</math> for instance). 
Then the topology on the vector subspace <math>M</math> is generated by the restrictions <math>\mathcal{P}\big\vert_M = \{p\big\vert_M : p \in \mathcal{P}\}</math> of these seminorms to <math>M.</math> Furthermore, for any continuous seminorm <math>q : M \to \Reals</math> on <math>M,</math> there exists some <math>p \in \mathcal{P}</math> such that <math>q \leq p.</math> --> 

{{Math proof|title=Proof of the [[#Hahn–Banach continuous extension theorem|continuous extension theorem]] for locally convex spaces{{sfn|Narici|Beckenstein|2011|p=182}}{{anchor|Proof of the continuous extension theorem for locally convex topological vector spaces}}|drop=hidden|proof=
Let <math>f</math> be a continuous linear functional defined on a vector subspace <math>M</math> of a [[locally convex topological vector space]] <math>X.</math> 
Because <math>X</math> is locally convex, there exists a continuous seminorm <math>p : X \to \Reals</math> on <math>X</math> that [[#dominated complex functional|dominates]] <math>f</math> (meaning that <math>|f(m)| \leq p(m)</math> for all <math>m \in M</math>). 
By the [[#Hahn–Banach theorem for real or complex vector spaces|Hahn–Banach theorem]], there exists a linear extension of <math>f</math> to <math>X,</math> call it <math>F,</math> that satisfies <math>|F| \leq p</math> on <math>X.</math> 
This linear functional <math>F</math> is continuous since <math>|F| \leq p</math> and <math>p</math> is a continuous seminorm.
}}

'''Proof for normed spaces'''

A linear functional <math>f</math> on a [[normed space]] is [[Continuous linear functional|continuous]] if and only if it is [[Bounded linear functional|bounded]], which means that its [[dual norm]] 
<math display=block>\|f\| = \sup \{|f(m)| : \|m\| \leq 1, m \in \operatorname{domain} f\}</math>
is finite, in which case <math>|f(m)| \leq \|f\| \|m\|</math> holds for every point <math>m</math> in its domain. 
Moreover, if <math>c \geq 0</math> is such that <math>|f(m)| \leq c \|m\|</math> for all <math>m</math> in the functional's domain, then necessarily <math>\|f\| \leq c.</math>
If <math>F</math> is a linear extension of a linear functional <math>f</math> then their dual norms always satisfy <math>\|f\| \leq \|F\|</math><ref group=proof name="ProofNormOfExtensionIsLarger" /> 
<!--(because <math display=inline>\{|f(m)| : \|m\| \leq 1, m \in \operatorname{domain} f\} \subseteq \{|F(x)| : \|x\| \leq 1, x \in \operatorname{domain} F\}</math>) -->
so that equality <math>\|f\| = \|F\|</math> is equivalent to <math>\|F\| \leq \|f\|,</math> which holds if and only if <math>|F(x)| \leq \|f\| \|x\|</math> for every point <math>x</math> in the extension's domain. 
This can be restated in terms of the function <math>\|f\| \, \|\cdot\| : X \to \Reals</math> defined by <math>x \mapsto \|f\| \, \|x\|,</math> which is always a [[seminorm]]:<ref group=note>Like every non-negative scalar multiple of a [[Norm (mathematics)|norm]], this seminorm <math>\|f\| \, \|\cdot\|</math> (the product of the non-negative real number <math>\|f\|</math> with the norm <math>\|\cdot\|</math>) is a norm when <math>\|f\|</math> is positive, although this fact is not needed for the proof.</ref>

:A linear extension of a [[bounded linear functional]] <math>f</math> is [[#norm-preserving linear extension|norm-preserving]] if and only if the extension is [[#dominated complex functional|dominated by]] the seminorm <math>\|f\| \, \|\cdot\|.</math>

Applying the [[#Hahn–Banach theorem for real or complex vector spaces|Hahn–Banach theorem]] to <math>f</math> with this seminorm <math>\|f\| \, \|\cdot\|</math> thus produces a dominated linear extension whose norm is (necessarily) equal to that of <math>f,</math> which proves the theorem:

{{Math proof|title=Proof of the [[#Norm-preserving Hahn–Banach continuous extension theorem|norm-preserving Hahn–Banach continuous extension theorem]]{{sfn|Narici|Beckenstein|2011|p=184}}|drop=hidden|proof=
Let <math>f</math> be a continuous linear functional defined on a vector subspace <math>M</math> of a normed space <math>X.</math> 
Then the function <math>p : X \to \Reals</math> defined by <math>p(x) = \|f\| \, \|x\|</math> is a seminorm on <math>X</math> that [[#dominated complex functional|dominates]] <math>f,</math> meaning that <math>|f(m)| \leq p(m)</math> holds for every <math>m \in M.</math>
By the [[#Hahn–Banach theorem for real or complex vector spaces|Hahn–Banach theorem]], there exists a linear functional <math>F</math> on <math>X</math> that extends <math>f</math> (which guarantees <math>\|f\| \leq \|F\|</math>) and that is also dominated by <math>p,</math> meaning that <math>|F(x)| \leq p(x)</math> for every <math>x \in X.</math> 
The fact that <math>\|f\|</math> is a real number such that <math>|F(x)| \leq \|f\| \|x\|</math> for every <math>x \in X,</math> guarantees <math>\|F\| \leq \|f\|.</math> 
Since <math>\|F\| = \|f\|</math> is finite, the linear functional <math>F</math> is bounded and thus continuous. 
}}

===Non-locally convex spaces===

The [[#Hahn–Banach continuous extension theorem|continuous extension theorem]] might fail if the [[topological vector space]] (TVS) <math>X</math> is not [[Locally convex topological vector space|locally convex]]. For example, for <math>0 < p < 1,</math> the [[Lp space|Lebesgue space]] <math>L^p([0, 1])</math> is a [[Complete metric space|complete]] [[Metrizable topological vector space|metrizable TVS]] (an [[F-space]]) that is {{em|not}} locally convex (in fact, its only convex open subsets are itself <math>L^p([0, 1])</math> and the empty set) and the only continuous linear functional on <math>L^p([0, 1])</math> is the constant <math>0</math> function {{harv|Rudin|1991|loc=§1.47}}. Since <math>L^p([0, 1])</math> is Hausdorff, every finite-dimensional vector subspace <math>M \subseteq L^p([0, 1])</math> is [[TVS-isomorphism|linearly homeomorphic]] to [[Euclidean space]] <math>\Reals^{\dim M}</math> or <math>\Complex^{\dim M}</math> (by [[F. Riesz's theorem]]) and so every non-zero linear functional <math>f</math> on <math>M</math> is continuous but none has a continuous linear extension to all of <math>L^p([0, 1]).</math> 
However, it is possible for a TVS <math>X</math> to not be locally convex but nevertheless have enough continuous linear functionals that its [[continuous dual space]] <math>X^*</math> [[Separating set|separates points]]; for such a TVS, a continuous linear functional defined on a vector subspace {{em|might}} have a continuous linear extension to the whole space. 

If the [[topological vector space|TVS]] <math>X</math> is not [[Locally convex topological vector space|locally convex]] then there might not exist any continuous seminorm <math>p : X \to \R</math> {{em|defined on <math>X</math>}} (not just on <math>M</math>) that dominates <math>f,</math> in which case the Hahn–Banach theorem can not be applied as it was in [[#Proof of the continuous extension theorem for locally convex topological vector spaces|the above proof]] of the continuous extension theorem. 
However, the proof's argument can be generalized to give a characterization of when a continuous linear functional has a continuous linear extension: If <math>X</math> is any TVS (not necessarily locally convex), then a continuous linear functional <math>f</math> defined on a vector subspace <math>M</math> has a continuous linear extension <math>F</math> to all of <math>X</math> if and only if there exists some continuous seminorm <math>p</math> on <math>X</math> that [[#dominated complex functional|dominates]] <math>f.</math>  Specifically, if given a continuous linear extension <math>F</math> then <math>p := |F|</math> is a continuous seminorm on <math>X</math> that dominates <math>f;</math> and conversely, if given a continuous seminorm <math>p : X \to \Reals</math> on <math>X</math> that dominates <math>f</math> then any dominated linear extension of <math>f</math> to <math>X</math> (the existence of which is guaranteed by the Hahn–Banach theorem) will be a continuous linear extension.

==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>

==Applications==

The Hahn–Banach theorem is the first sign of an important philosophy in [[functional analysis]]: to understand a space, one should understand its [[continuous functional]]s.

For example, linear subspaces are characterized by functionals: if {{mvar|X}} is a normed vector space with linear subspace {{mvar|M}} (not necessarily closed) and if <math>z</math> is an element of {{mvar|X}} not in the [[Closure (topology)|closure]] of {{mvar|M}}, then there exists a continuous linear map <math>f : X \to \mathbf{K}</math> with <math>f(m) = 0</math> for all <math>m \in M,</math> <math>f(z) = 1,</math> and <math>\|f\| = \operatorname{dist}(z, M)^{-1}.</math>  (To see this, note that <math>\operatorname{dist}(\cdot, M)</math> is a sublinear function.)  Moreover, if <math>z</math> is an element of {{mvar|X}}, then there exists a continuous linear map <math>f : X \to \mathbf{K}</math> such that <math>f(z) = \|z\|</math> and <math>\|f\| \leq 1.</math>  This implies that the [[Reflexive space#Normed spaces|natural injection]] <math>J</math> from a normed space {{mvar|X}} into its [[Reflexive space#Normed spaces|double dual]] <math>V^{**}</math> is isometric.

That last result also suggests that the Hahn–Banach theorem can often be used to locate a "nicer" topology in which to work.  For example, many results in functional analysis assume that a space is [[Hausdorff space|Hausdorff]] or [[Locally convex space|locally convex]].  However, suppose {{mvar|X}} is a topological vector space, not necessarily Hausdorff or [[Locally convex topological vector space|locally convex]], but with a nonempty, proper, convex, open set {{mvar|M}}.  Then geometric Hahn–Banach implies that there is a hyperplane separating {{mvar|M}} from any other point.  In particular, there must exist a nonzero functional on {{mvar|X}} — that is, the [[Dual space#Continuous dual space|continuous dual space]] <math>X^*</math> is non-trivial.{{sfn|Narici|Beckenstein|2011|pp=177-220}}{{sfn|Schaefer|Wolff|1999|p=47}} Considering {{mvar|X}} with the [[weak topology]] induced by <math>X^*,</math> then {{mvar|X}} becomes locally convex; by the second bullet of geometric Hahn–Banach, the weak topology on this new space [[separates points]]. 
Thus {{mvar|X}} with this weak topology becomes [[Hausdorff space|Hausdorff]].  This sometimes allows some results from locally convex topological vector spaces to be applied to non-Hausdorff and non-locally convex spaces.

===Partial differential equations===

The Hahn–Banach theorem is often useful when one wishes to apply the method of [[a priori estimate]]s.  Suppose that we wish to solve the linear differential equation <math>P u = f</math> for <math>u,</math> with <math>f</math> given in some Banach space {{mvar|X}}.  If we have control on the size of <math>u</math> in terms of <math>\|f\|_X</math> and we can think of <math>u</math> as a bounded linear functional on some suitable space of test functions <math>g,</math> then we can view <math>f</math> as a linear functional by adjunction: <math>(f, g) = (u, P^*g).</math>  At first, this functional is only defined on the image of <math>P,</math> but using the Hahn–Banach theorem, we can try to extend it to the entire codomain {{mvar|X}}.  The resulting functional is often defined to be a [[Weak solution|weak solution to the equation]].

===Characterizing reflexive Banach spaces===

{{Math theorem
| name = Theorem{{sfn|Narici|Beckenstein|2011|p=212}}
| math_statement = A real Banach space is [[Reflexive space|reflexive]] if and only if every pair of non-empty disjoint closed convex subsets, one of which is bounded, can be strictly separated by a hyperplane.
}}

===Example from Fredholm theory===
To illustrate an actual application of the Hahn–Banach theorem, we will now prove a result that follows almost entirely from the Hahn–Banach theorem.

{{Math theorem
| name = Proposition
| math_statement = Suppose <math>X</math> is a Hausdorff locally convex TVS over the field <math>\mathbf{K}</math> and <math>Y</math> is a vector subspace of <math>X</math> that is [[TVS isomorphism|TVS–isomorphic]] to <math>\mathbf{K}^I</math> for some set <math>I.</math> 
Then <math>Y</math> is a closed and [[Complemented subspace|complemented]] vector subspace of <math>X.</math>
}}

{{Math proof|drop=hidden|proof=
Since <math>\mathbf{K}^I</math> is a complete TVS so is <math>Y,</math> and since any complete subset of a Hausdorff TVS is closed, <math>Y</math> is a closed subset of <math>X.</math> 
Let <math>f = \left(f_i\right)_{i \in I} : Y \to \mathbf{K}^I</math> be a TVS isomorphism, so that each <math>f_i : Y \to \mathbf{K}</math> is a continuous surjective linear functional. 
By the Hahn–Banach theorem, we may extend each <math>f_i</math> to a continuous linear functional <math>F_i : X \to \mathbf{K}</math> on <math>X.</math> 
Let <math>F := \left(F_i\right)_{i \in I} : X \to \mathbf{K}^I</math> so <math>F</math> is a continuous linear surjection such that its restriction to <math>Y</math> is <math>F\big\vert_Y = \left(F_i\big\vert_Y\right)_{i \in I} = \left(f_i\right)_{i \in I} = f.</math> 
Let <math>P := f^{-1} \circ F : X \to Y,</math> which is a continuous linear map whose restriction to <math>Y</math> is <math>P\big\vert_Y = f^{-1} \circ F\big\vert_Y = f^{-1} \circ f = \mathbf{1}_Y,</math> where <math>\mathbb{1}_Y</math> denotes the [[identity map]] on <math>Y.</math> 
This shows that <math>P</math> is a continuous [[linear projection]] onto <math>Y</math> (that is, <math>P \circ P = P</math>). 
Thus <math>Y</math> is complemented in <math>X</math> and <math>X = Y \oplus \ker P</math> in the category of TVSs. <math>\blacksquare</math>
}}

The above result may be used to show that every closed vector subspace of <math>\R^{\N}</math> is complemented because any such space is either finite dimensional or else TVS–isomorphic to <math>\R^{\N}.</math>

==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>
}}

==Converse==
Let {{mvar|X}} be a topological vector space.  A vector subspace {{mvar|M}} of {{mvar|X}} has '''the extension property''' if any continuous linear functional on {{mvar|M}} can be extended to a continuous linear functional on {{mvar|X}}, and we say that {{mvar|X}} has the '''Hahn–Banach extension property''' ('''HBEP''') if every vector subspace of {{mvar|X}} has the extension property.{{sfn|Narici|Beckenstein|2011|pp=225-273}}

The Hahn–Banach theorem guarantees that every Hausdorff locally convex space has the HBEP.  For complete [[metrizable TVS|metrizable topological vector space]]s there is a converse, due to Kalton: every complete metrizable TVS with the Hahn–Banach extension property is locally convex.{{sfn|Narici|Beckenstein|2011|pp=225-273}}  On the other hand, a vector space {{mvar|X}} of uncountable dimension, endowed with the [[finest vector topology]], then this is a topological vector spaces with the Hahn–Banach extension property that is neither locally convex nor metrizable.{{sfn|Narici|Beckenstein|2011|pp=225-273}}

A vector subspace {{mvar|M}} of a TVS {{mvar|X}} has '''the separation property''' if for every element of {{mvar|X}} such that <math>x \not\in M,</math> there exists a continuous linear functional <math>f</math> on {{mvar|X}} such that <math>f(x) \neq 0</math> and <math>f(m) = 0</math> for all <math>m \in M.</math>  Clearly, the continuous dual space of a TVS {{mvar|X}} separates points on {{mvar|X}} if and only if <math>\{0\},</math> has the separation property.  In 1992, Kakol proved that any infinite dimensional vector space {{mvar|X}}, there exist TVS-topologies on {{mvar|X}} that do not have the HBEP despite having enough continuous linear functionals for the continuous dual space to separate points on {{mvar|X}}.  However, if {{mvar|X}} is a TVS then {{em|every}} vector subspace of {{mvar|X}} has the extension property if and only if {{em|every}} vector subspace of {{mvar|X}} has the separation property.{{sfn|Narici|Beckenstein|2011|pp=225-273}}

==Relation to axiom of choice and other theorems==
{{See also|Krein–Milman theorem#Relation to other statements}}

{{anchor|Relation to axiom of choice}}The proof of the [[#Hahn–Banach dominated extension theorem|Hahn–Banach theorem for real vector spaces]] ('''HB''') commonly uses [[Zorn's lemma]], which in the axiomatic framework of [[Zermelo–Fraenkel set theory]] ('''ZF''') is equivalent to the [[axiom of choice]] ('''AC''').  It was discovered by [[Jerzy Łoś|Łoś]] and [[Czesław Ryll-Nardzewski|Ryll-Nardzewski]]{{sfn|Łoś|Ryll-Nardzewski|1951|pp=233–237}} and independently by [[Wilhelmus Luxemburg|Luxemburg]]{{sfn|Luxemburg|1962|p=}} that '''HB''' can be proved using the [[ultrafilter lemma]] ('''UL'''), which is equivalent (under '''ZF''') to the [[Boolean prime ideal theorem]] ('''BPI'''). '''BPI''' is strictly weaker than the axiom of choice and it was later shown that '''HB''' is strictly weaker than '''BPI'''.{{sfn|Pincus|1974|pp=203–205}}

The [[ultrafilter lemma]] is equivalent (under '''ZF''') to the [[Banach–Alaoglu theorem]],{{sfn|Schechter|1996|pp=766–767}} which is another foundational theorem in [[functional analysis]]. Although the Banach–Alaoglu theorem implies '''HB''',<ref name="Muger2020">{{cite book|last=Muger|first= Michael|title=Topology for the Working Mathematician|year=2020}}</ref> it is not equivalent to it (said differently, the Banach–Alaoglu theorem is strictly stronger than '''HB'''). 
However, '''HB''' is equivalent to [[Banach–Alaoglu theorem#Relation to the Hahn–Banach theorem|a certain weakened version of the Banach–Alaoglu theorem]] for normed spaces.<ref name=BellFremlin1972>{{cite journal|last1=Bell|first1=J.|last2=Fremlin|first2=David|title=A Geometric Form of the Axiom of Choice|journal=[[Fundamenta Mathematicae]]|date=1972|volume=77|issue=2|pages=167–170|doi=10.4064/fm-77-2-167-170|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm77/fm77116.pdf|access-date=26 Dec 2021}}</ref> 
The Hahn–Banach theorem is also equivalent to the following statement:<ref>{{cite book|last=Schechter|first=Eric|title=Handbook of Analysis and its Foundations|page=620|author-link=Eric Schechter}}</ref>

:(∗): On every [[Boolean algebra (structure)|Boolean algebra]] {{mvar|B}} there exists a "probability charge", that is: a non-constant finitely additive map from <math>B</math> into <math>[0, 1].</math>

('''BPI''' is equivalent to the statement that there are always non-constant probability charges which take only the values 0 and 1.)

In '''ZF''', the Hahn–Banach theorem suffices to derive the existence of a non-Lebesgue measurable set.<ref>{{cite journal|last1=Foreman|first1=M.|last2=Wehrung|first2=F.|year=1991|title=The Hahn–Banach theorem implies the existence of a non-Lebesgue measurable set|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm138/fm13812.pdf|journal=Fundamenta Mathematicae|volume=138|pages=13–19|doi=10.4064/fm-138-1-13-19|doi-access=free}}</ref>  Moreover, the Hahn–Banach theorem implies the [[Banach–Tarski paradox]].<ref>{{cite journal|last=Pawlikowski|first=Janusz|year=1991|title=The Hahn–Banach theorem implies the Banach–Tarski paradox|journal=Fundamenta Mathematicae|volume=138|pages=21–22|doi=10.4064/fm-138-1-21-22|doi-access=free}}</ref>

For [[Separable space|separable]] [[Banach space]]s, D. K. Brown and S. G. Simpson proved that the Hahn–Banach theorem follows from WKL<sub>0</sub>, a weak subsystem of [[second-order arithmetic]] that takes a form of [[Kőnig's lemma]] restricted to binary trees as an axiom.  In fact, they prove that under a weak set of assumptions, the two are equivalent, an example of [[reverse mathematics]].<ref>{{cite journal|last1=Brown|first1=D. K.|last2=Simpson|first2=S. G.|year=1986|title=Which set existence axioms are needed to prove the separable Hahn–Banach theorem?|journal=[[Annals of Pure and Applied Logic]]|volume=31|pages=123–144|doi=10.1016/0168-0072(86)90066-7 |doi-access=}} [http://www.math.psu.edu/simpson/papers/hilbert/node7.html#3 Source of citation].</ref><ref>Simpson, Stephen G. (2009), Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, {{ISBN|978-0-521-88439-6}}, {{MR|2517689}}</ref>

==See also==

* {{annotated link|Farkas' lemma}}
* {{annotated link|Fichera's existence principle}}
* {{annotated link|M. Riesz extension theorem}}
* {{annotated link|Separating axis theorem}}
* {{annotated link|Vector-valued Hahn–Banach theorems}}

==Notes==

{{reflist|group=note|refs=
* <ref group=note name="GeometricIllustration">Geometric illustration:
The geometric idea of the above proof can be fully presented in the case of <math>X = \R^2, M = \{(x, 0) : x \in \R\}.</math>

First, define the simple-minded extension <math>f_0(x, y) = f(x),</math> It doesn't work, since maybe <math>f_0 \leq p</math>. But it is a step in the right direction. <math>p-f_0</math> is still convex, and <math>p-f_0 \geq f-f_0.</math> Further, <math>f-f_0</math> is identically zero on the x-axis. Thus we have reduced to the case of <math>f = 0, p \geq 0</math> on the x-axis.

If <math>p \geq 0</math> on <math>\R^2,</math> then we are done. Otherwise, pick some <math>v \in \R^2,</math> such that <math>p(v) < 0.</math>

The idea now is to perform a simultaneous bounding of <math>p</math> on <math>v + M</math> and <math>-v+M</math> such that <math>p \geq b</math> on <math>v+M</math> and <math>p \geq -b</math> on <math>-v+M,</math> then defining <math>\tilde f(w + rv) = rb</math> would give the desired extension.

Since <math>-v+M, v+M</math> are on opposite sides of <math>M,</math> and <math>p < 0</math> at some point on <math>v+M,</math> by convexity of <math>p,</math> we must have <math>p \geq 0</math> on all points on <math>-v+M.</math> Thus <math>\inf_{u\in -v + M} p(u)</math> is finite.

Geometrically, this works because <math>\{z : p(z) < 0\}</math> is a convex set that is disjoint from <math>M,</math> and thus must lie entirely on one side of <math>M.</math>

Define <math>b = -\inf_{u\in -v + M} p(u).</math> This satisfies <math>p\geq -b</math> on <math>-v+M.</math> It remains to check the other side.

For all <math>v+w \in v+M,</math> convexity implies that for all <math>-v+w' \in -v+M, p(v+w) + p(-v +w') \geq 2p((w+w')/2) = 0,</math> thus 
<math>p(v + w) \geq \sup_{u\in -v + M} -p(u) = b.</math>

Since during the proof, we only used convexity of <math>p</math>, we see that the lemma remains true for merely convex <math>p.</math>
</ref>
}}

'''Proofs'''

{{reflist|group=proof|refs=
* <ref group=proof name="ProofNormOfExtensionIsLarger">The map <math>F</math> being an extension of <math>f</math> means that <math>\operatorname{domain} f \subseteq \operatorname{domain} F</math> and <math>F(m) = f(m)</math> for every <math>m \in \operatorname{domain} f.</math> Consequently,
<math display=block>\{|f(m)| : \|m\| \leq 1, m \in \operatorname{domain} f\} = \{|F(m)|: \|m\| \leq 1, m \in \operatorname{domain} f\} \subseteq \{|F(x)\,| : \|x\| \leq 1, x \in \operatorname{domain} F\}</math> and so the [[supremum]] of the set on the left hand side, which is <math>\|f\|,</math> does not exceed the supremum of the right hand side, which is <math>\|F\|.</math> In other words, <math>\|f\| \leq \|F\|.</math>
</ref>
}}

==References==

{{reflist|30em}}

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