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{{short description|Theorem on extension of bounded linear functionals}}
The '''Hahn–Banach theorem''' is a central tool in [[functional analysis]]. 
It allows the extension of [[Bounded operator|bounded linear functionals]] defined on a subspace of some [[vector space]] to the whole space, and it also shows that there are "enough" [[Continuous function (topology)|continuous]] linear functionals defined on every [[normed vector space]] to make the study of the [[dual space]] "interesting". 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=Gȧrding|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 [[1921]] 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 spaces 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 <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 or 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> 
Hence the reason why 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 sublinear,{{citation needed|date=February 2023}} 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|volume=12|isbn=978-90-481-4661-1|url=https://www.researchgate.net/publication/240011075_Fundamentals_of_Functional_Analysis|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> be 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]] {{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=free}} [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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* [[Terence Tao|Tao, Terence]], [http://terrytao.wordpress.com/2007/11/30/the-hahn-banach-theorem-mengers-theorem-and-hellys-theorem The Hahn–Banach theorem, Menger's theorem, and Helly's theorem]
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* {{springer|title=Hahn–Banach theorem|id=p/h046130}}

{{Functional Analysis}}
{{TopologicalVectorSpaces}}
{{Banach spaces}}

{{DEFAULTSORT:Hahn-Banach theorem}}

[[Category:Articles containing proofs]]
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[[Category:Linear functionals]]
[[Category:Theorems in functional analysis]]
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