Hahn–Banach theorem

The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s. The special case of the theorem for the space ${\displaystyle C[a,b]}$ of continuous functions on an interval was proved earlier (in 1912) by Eduard Helly,[1] 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.[2]

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 (${\displaystyle \mathbb {C} ^{\mathbb {N} }}$) had an extension of the same norm. Helly did this through the technique of first proving that a one-dimensional extension exists (where the linear functional has its domain extended by one dimension) and then using 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 functions. Whereas Helly's proof used mathematical induction, Hahn and Banach both used transfinite induction.[3]

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 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 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 ${\displaystyle L^{p}([0,1])}$ and ${\displaystyle C([a,b])}$ 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:[3]

(The vector problem) Given a collection ${\displaystyle \left(f_{i}\right)_{i\in I}}$ of bounded linear functionals on a normed space ${\displaystyle X}$ and a collection of scalars ${\displaystyle \left(c_{i}\right)_{i\in I},}$ determine if there is an ${\displaystyle x\in X}$ such that ${\displaystyle f_{i}(x)=c_{i}}$ for all ${\displaystyle i\in I.}$

If ${\displaystyle X}$ happens to be a reflexive space then to solve the vector problem, it suffices to solve the following dual problem:[3]

(The functional problem) Given a collection ${\displaystyle \left(x_{i}\right)_{i\in I}}$ of vectors in a normed space ${\displaystyle X}$ and a collection of scalars ${\displaystyle \left(c_{i}\right)_{i\in I},}$ determine if there is a bounded linear functional ${\displaystyle f}$ on ${\displaystyle X}$ such that ${\displaystyle f\left(x_{i}\right)=c_{i}}$ for all ${\displaystyle i\in I.}$

Riesz went on to define ${\displaystyle L^{p}([0,1])}$ space (${\displaystyle 1<p<\infty }$) in 1910 and the ${\displaystyle \ell ^{p}}$ 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.[3]

Theorem[3] (The functional problem) — Let ${\displaystyle \left(x_{i}\right)_{i\in I}}$ be vectors in a real or complex normed space ${\displaystyle X}$ and let ${\displaystyle \left(c_{i}\right)_{i\in I}}$ be scalars also indexed by ${\displaystyle I\neq \varnothing .}$

There exists a continuous linear functional ${\displaystyle f}$ on ${\displaystyle X}$ such that ${\displaystyle f\left(x_{i}\right)=c_{i}}$ for all ${\displaystyle i\in I}$ if and only if there exists a ${\displaystyle K>0}$ such that for any choice of scalars ${\displaystyle \left(s_{i}\right)_{i\in I}}$ where all but finitely many ${\displaystyle s_{i}}$ are ${\displaystyle 0,}$ the following holds:

$${\displaystyle \left|\sum _{i\in I}s_{i}c_{i}\right|\leq K\left\|\sum _{i\in I}s_{i}x_{i}\right\|.}$$

The Hahn–Banach theorem can be deduced from the above theorem.[3] If ${\displaystyle X}$ is reflexive then this theorem solves the vector problem.

A real-valued function ${\displaystyle f:M\to \mathbb {R} }$ defined on a subset ${\displaystyle M}$ of ${\displaystyle X}$ is said to be dominated (above) by a function ${\displaystyle p:X\to \mathbb {R} }$ if ${\displaystyle f(m)\leq p(m)}$ for every ${\displaystyle m\in M.}$ Hence the reason why the following version of the Hahn-Banach theorem is called the dominated extension theorem.

Hahn–Banach dominated extension theorem (for real linear functionals)[4][5][6] — If ${\displaystyle p:X\to \mathbb {R} }$ is a sublinear function (such as a norm or seminorm for example) defined on a real vector space ${\displaystyle X}$ then any linear functional defined on a vector subspace of ${\displaystyle X}$ that is dominated above by ${\displaystyle p}$ has at least one linear extension to all of ${\displaystyle X}$ that is also dominated above by ${\displaystyle p.}$

Explicitly, if ${\displaystyle p:X\to \mathbb {R} }$ is a sublinear function, which by definition means that it satisfies

$${\displaystyle p(x+y)\leq p(x)+p(y)\quad {\text{ and }}\quad p(tx)=tp(x)\qquad {\text{ for all }}\;x,y\in X\;{\text{ and all real }}\;t\geq 0,}$$

and if ${\displaystyle f:M\to \mathbb {R} }$ is a linear functional defined on a vector subspace ${\displaystyle M}$ of ${\displaystyle X}$ such that

$${\displaystyle f(m)\leq p(m)\quad {\text{ for all }}m\in M}$$

then there exists a linear functional ${\displaystyle F:X\to \mathbb {R} }$ such that

$${\displaystyle F(m)=f(m)\quad {\text{ for all }}m\in M,}$$

$${\displaystyle F(x)\leq p(x)\quad ~\;\,{\text{ for all }}x\in X.}$$

Moreover, if ${\displaystyle p}$ is a seminorm then ${\displaystyle |F(x)|\leq p(x)}$ necessarily holds for all ${\displaystyle x\in X.}$

The theorem remains true if the requirements on ${\displaystyle p}$ are relaxed to require only that ${\displaystyle p}$ be a convex function:[7][8]

$${\displaystyle p(tx+(1-t)y)\leq tp(x)+(1-t)p(y)\qquad {\text{ for all }}0<t<1{\text{ and }}x,y\in X.}$$

A function ${\displaystyle p:X\to \mathbb {R} }$ is convex and satisfies ${\displaystyle p(0)\leq 0}$ if and only if ${\displaystyle p(ax+by)\leq ap(x)+bp(y)}$ for all vectors ${\displaystyle x,y\in X}$ and all non-negative real ${\displaystyle a,b\geq 0}$ such that ${\displaystyle a+b\leq 1.}$ Every sublinear function is a convex function. On the other hand, if ${\displaystyle p:X\to \mathbb {R} }$ is convex, the function ${\displaystyle p_{0}\leq p}$ defined by ${\displaystyle p_{0}(x):=\inf _{t>0}{\frac {p(tx)}{t}}}$ is sublinear, and satisfies ${\displaystyle F\leq p_{0}}$ for all linear functional ${\displaystyle F\leq p}$. So the extension of the Hahn-Banach theorem to convex functionals is easy, but has not a much larger content than the classical one stated for sublinear functionals.

If ${\displaystyle F:X\to \mathbb {R} }$ is linear then ${\displaystyle F\leq p}$ if and only if[4]

$${\displaystyle -p(-x)\leq F(x)\leq p(x)\quad {\text{ for all }}x\in X,}$$

which is the (equivalent) conclusion that some authors[4] write instead of ${\displaystyle F\leq p.}$ It follows that if ${\displaystyle p:X\to \mathbb {R} }$ is also symmetric, meaning that ${\displaystyle p(-x)=p(x)}$ holds for all ${\displaystyle x\in X,}$ then ${\displaystyle F\leq p}$ if and only ${\displaystyle |F|\leq p.}$ Every norm is a seminorm and both are symmetric 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 ${\displaystyle \mathbb {R} \to \mathbb {R} }$ on ${\displaystyle X:=\mathbb {R} }$ 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.

Hahn–Banach theorem[3][9] — Suppose ${\displaystyle p:X\to \mathbb {R} }$ a seminorm on a vector space ${\displaystyle X}$ over the field ${\displaystyle \mathbf {K} ,}$ which is either ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} .}$ If ${\displaystyle f:M\to \mathbf {K} }$ is a linear functional on a vector subspace ${\displaystyle M}$ such that

$${\displaystyle |f(m)|\leq p(m)\quad {\text{ for all }}m\in M.}$$

then there exists a linear functional ${\displaystyle F:X\to \mathbf {K} }$ such that

$${\displaystyle F(m)=f(m)\quad \;{\text{ for all }}m\in M,}$$

$${\displaystyle |F(x)|\leq p(x)\quad \;\,{\text{ for all }}x\in X.}$$

The theorem remains true if the requirements on ${\displaystyle p}$ are relaxed to require only that for all ${\displaystyle x,y\in X}$ and all scalars ${\displaystyle a}$ and ${\displaystyle b}$ satisfying ${\displaystyle |a|+|b|\leq 1,}$[8]

$${\displaystyle p(ax+by)\leq |a|p(x)+|b|p(y).}$$

This condition holds if and only if ${\displaystyle p}$ is a convex and balanced function satisfying ${\displaystyle p(0)\leq 0,}$ or equivalently, if and only if it is convex, satisfies ${\displaystyle p(0)\leq 0,}$ and ${\displaystyle p(ux)\leq p(x)}$ for all ${\displaystyle x\in X}$ and all unit length scalars ${\displaystyle u.}$

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

Hahn–Banach dominated extension theorem: If ${\displaystyle p:X\to \mathbb {R} }$ is a seminorm defined on a real or complex vector space ${\displaystyle X,}$ then every dominated linear functional defined on a vector subspace of ${\displaystyle X}$ has a dominated linear extension to all of ${\displaystyle X.}$ In the case where ${\displaystyle X}$ is a real vector space and ${\displaystyle p:X\to \mathbb {R} }$ is merely a convex or sublinear function, this conclusion will remain true if both instances of "dominated" (meaning ${\displaystyle |F|\leq p}$) are weakened to instead mean "dominated above" (meaning ${\displaystyle F\leq p}$).[7][8]

Proof

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

Every linear functional ${\displaystyle F:X\to \mathbb {C} }$ on a complex vector space is completely determined by its real part ${\displaystyle \;\operatorname {Re} F:X\to \mathbb {R} \;}$ through the formula[6][proof 1]

$${\displaystyle F(x)\;=\;\operatorname {Re} F(x)-i\operatorname {Re} F(ix)\qquad {\text{ for all }}x\in X}$$

and moreover, if ${\displaystyle \|\cdot \|}$ is a norm on ${\displaystyle X}$ then their operator norms are equal: ${\displaystyle \|F\|=\|\operatorname {Re} F\|.}$[10] In particular, a linear functional on ${\displaystyle X}$ extends another one defined on ${\displaystyle M\subseteq X}$ if and only if their real parts are equal on ${\displaystyle M}$ (in other words, a linear functional ${\displaystyle F}$ extends ${\displaystyle f}$ if and only if ${\displaystyle \operatorname {Re} F}$ extends ${\displaystyle \operatorname {Re} f}$). The real part of a linear functional on ${\displaystyle X}$ is always a real-linear functional (meaning that it is linear when ${\displaystyle X}$ is considered as a real vector space) and if ${\displaystyle R:X\to \mathbb {R} }$ is a real-linear functional on a complex vector space then ${\displaystyle x\mapsto R(x)-iR(ix)}$ defines the unique linear functional on ${\displaystyle X}$ whose real part is ${\displaystyle R.}$

If ${\displaystyle F}$ is a linear functional on a (complex or real) vector space ${\displaystyle X}$ and if ${\displaystyle p:X\to \mathbb {R} }$ is a seminorm then[6][proof 2]

$${\displaystyle |F|\,\leq \,p\quad {\text{ if and only if }}\quad \operatorname {Re} F\,\leq \,p.}$$

Stated in simpler language, a linear functional is dominated by a seminorm ${\displaystyle p}$ if and only if its real part is dominated above by ${\displaystyle p.}$

Proof of Hahn–Banach for complex vector spaces by reduction to real vector spaces[3]

Suppose ${\displaystyle p:X\to \mathbb {R} }$ is a seminorm on a complex vector space ${\displaystyle X}$ and let ${\displaystyle f:M\to \mathbb {C} }$ be a linear functional defined on a vector subspace ${\displaystyle M}$ of ${\displaystyle X}$ that satisfies ${\displaystyle |f|\leq p}$ on ${\displaystyle M.}$ Consider ${\displaystyle X}$ as a real vector space and apply the Hahn–Banach theorem for real vector spaces to the real-linear functional ${\displaystyle \;\operatorname {Re} f:M\to \mathbb {R} \;}$ to obtain a real-linear extension ${\displaystyle R:X\to \mathbb {R} }$ that is also dominated above by ${\displaystyle p,}$ which means that it satisfies ${\displaystyle R\leq p}$ on ${\displaystyle X}$ and ${\displaystyle R=\operatorname {Re} f}$ on ${\displaystyle M.}$ The map ${\displaystyle F:X\to \mathbb {C} }$ defined by ${\displaystyle F(x)\;=\;R(x)-iR(ix)}$ is a linear functional on ${\displaystyle X}$ that extends ${\displaystyle f}$ (because their real parts agree on ${\displaystyle M}$) and satisfies ${\displaystyle |F|\leq p}$ on ${\displaystyle X}$ (because ${\displaystyle \operatorname {Re} F\leq p}$ and ${\displaystyle p}$ is a seminorm). ${\displaystyle \blacksquare }$

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

Continuity

A linear functional ${\displaystyle F}$ on a topological vector space is continuous if and only if this is true of its real part ${\displaystyle \operatorname {Re} F;}$ if the domain is a normed space then ${\displaystyle \|F\|=\|\operatorname {Re} F\|}$ (where one side is infinite if and only if the other side is infinite).[10] Assume ${\displaystyle X}$ is a topological vector space and ${\displaystyle p:X\to \mathbb {R} }$ is sublinear function. If ${\displaystyle p}$ is a continuous sublinear function that dominates a linear functional ${\displaystyle F}$ then ${\displaystyle F}$ is necessarily continuous.[6] Moreover, a linear functional ${\displaystyle F}$ is continuous if and only if its absolute value ${\displaystyle |F|}$ (which is a seminorm that dominates ${\displaystyle F}$) is continuous.[6] In particular, a linear functional is continuous if and only if it is dominated by some continuous sublinear function.

Proof

The 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 ${\displaystyle M}$ to a larger vector space in which ${\displaystyle M}$ has codimension ${\displaystyle 1.}$[3]

Lemma[6] (One–dimensional dominated extension theorem) — Let ${\displaystyle p:X\to \mathbb {R} }$ be a sublinear function on a real vector space ${\displaystyle X,}$ let ${\displaystyle f:M\to \mathbb {R} }$ a linear functional on a proper vector subspace ${\displaystyle M\subsetneq X}$ such that ${\displaystyle f\leq p}$ on ${\displaystyle M}$ (meaning ${\displaystyle f(m)\leq p(m)}$ for all ${\displaystyle m\in M}$), and let ${\displaystyle x\in X}$ be a vector not in ${\displaystyle M}$ (so ${\displaystyle M\oplus \mathbb {R} x=\operatorname {span} \{M,x\}}$). There exists a linear extension ${\displaystyle F:M\oplus \mathbb {R} x\to \mathbb {R} }$ of ${\displaystyle f}$ such that ${\displaystyle F\leq p}$ on ${\displaystyle M\oplus \mathbb {R} x.}$

Proof[6]

Given any real number ${\displaystyle b,}$ the map ${\displaystyle F_{b}:M\oplus \mathbb {R} x\to \mathbb {R} }$ defined by ${\displaystyle F_{b}(m+rx)=f(m)+rb}$ is always a linear extension of ${\displaystyle f}$ to ${\displaystyle M\oplus \mathbb {R} x}$[note 1] but it might not satisfy ${\displaystyle F_{b}\leq p.}$ It will be shown that ${\displaystyle b}$ can always be chosen so as to guarantee that ${\displaystyle F_{b}\leq p,}$ which will complete the proof.

If ${\displaystyle m,n\in M}$ then

$${\displaystyle f(m)-f(n)=f(m-n)\leq p(m-n)=p(m+x-x-n)\leq p(m+x)+p(-x-n)}$$

which implies

$${\displaystyle -p(-n-x)-f(n)~\leq ~p(m+x)-f(m).}$$

So define

$${\displaystyle a=\sup _{n\in M}[-p(-n-x)-f(n)]\qquad {\text{ and }}\qquad c=\inf _{m\in M}[p(m+x)-f(m)]}$$

where ${\displaystyle a\leq c}$ are real numbers. To guarantee ${\displaystyle F_{b}\leq p,}$ it suffices that ${\displaystyle a\leq b\leq c}$ (in fact, this is also necessary[note 2]) because then ${\displaystyle b}$ satisfies "the decisive inequality"[6]

$${\displaystyle -p(-n-x)-f(n)~\leq ~b~\leq ~p(m+x)-f(m)\qquad {\text{ for all }}\;m,n\in M.}$$

To see that ${\displaystyle f(m)+rb\leq p(m+rx)}$ follows,[note 3] assume ${\displaystyle r\neq 0}$ and substitute ${\displaystyle {\tfrac {1}{r}}m}$ in for both ${\displaystyle m}$ and ${\displaystyle n}$ to obtain

$${\displaystyle -p\left(-{\tfrac {1}{r}}m-x\right)-{\tfrac {1}{r}}f\left(m\right)~\leq ~b~\leq ~p\left({\tfrac {1}{r}}m+x\right)-{\tfrac {1}{r}}f\left(m\right).}$$

If ${\displaystyle r>0}$ (respectively, if ${\displaystyle r<0}$) then the right (respectively, the left) hand side equals ${\displaystyle {\tfrac {1}{r}}\left[p(m+rx)-f(m)\right]}$ so that multiplying by ${\displaystyle r}$ gives ${\displaystyle rb\leq p(m+rx)-f(m).}$ ${\displaystyle \blacksquare }$

This lemma remains true if ${\displaystyle p:X\to \mathbb {R} }$ is merely a convex function instead of a sublinear function.[7][8]

Proof

Assume that ${\displaystyle p}$ is convex, which means that ${\displaystyle p(ty+(1-t)z)\leq tp(y)+(1-t)p(z)}$ for all ${\displaystyle 0\leq t\leq 1}$ and ${\displaystyle y,z\in X.}$ Let ${\displaystyle M,}$ ${\displaystyle f:M\to \mathbb {R} ,}$ and ${\displaystyle x\in X\setminus M}$ be as in the lemma's statement. Given any ${\displaystyle m,n\in M}$ and any positive real ${\displaystyle r,s>0,}$ the positive real numbers ${\displaystyle t:={\tfrac {s}{r+s}}}$ and ${\displaystyle {\tfrac {r}{r+s}}=1-t}$ sum to ${\displaystyle 1}$ so that the convexity of ${\displaystyle p}$ on ${\displaystyle X}$ guarantees $${\displaystyle {\begin{alignedat}{9}p\left({\tfrac {s}{r+s}}m+{\tfrac {r}{r+s}}n\right)~&=~p{\big (}{\tfrac {s}{r+s}}(m-rx)&&+{\tfrac {r}{r+s}}(n+sx){\big )}&&\\&\leq ~{\tfrac {s}{r+s}}\;p(m-rx)&&+{\tfrac {r}{r+s}}\;p(n+sx)&&\\\end{alignedat}}}$$ and hence $${\displaystyle {\begin{alignedat}{9}sf(m)+rf(n)~&=~(r+s)\;f\left({\tfrac {s}{r+s}}m+{\tfrac {r}{r+s}}n\right)&&\qquad {\text{ by linearity of }}f\\&\leq ~(r+s)\;p\left({\tfrac {s}{r+s}}m+{\tfrac {r}{r+s}}n\right)&&\qquad f\leq p{\text{ on }}M\\&\leq ~sp(m-rx)+rp(n+sx)\\\end{alignedat}}}$$ thus proving that ${\displaystyle -sp(m-rx)+sf(m)~\leq ~rp(n+sx)-rf(n),}$ which after multiplying both sides by ${\displaystyle {\tfrac {1}{rs}}}$ becomes $${\displaystyle {\tfrac {1}{r}}[-p(m-rx)+f(m)]~\leq ~{\tfrac {1}{s}}[p(n+sx)-f(n)].}$$ This implies that the values defined by $${\displaystyle a=\sup _{\stackrel {m\in M}{r>0}}{\tfrac {1}{r}}[-p(m-rx)+f(m)]\qquad {\text{ and }}\qquad c=\inf _{\stackrel {n\in M}{s>0}}{\tfrac {1}{s}}[p(n+sx)-f(n)]}$$ are real numbers that satisfy ${\displaystyle a\leq c.}$ As in the above proof of the one–dimensional dominated extension theorem above, for any real ${\displaystyle b\in \mathbb {R} }$ define ${\displaystyle F_{b}:M\oplus \mathbb {R} x\to \mathbb {R} }$ by ${\displaystyle F_{b}(m+rx)=f(m)+rb.}$ It can be verified that if ${\displaystyle a\leq b\leq c}$ then ${\displaystyle F_{b}\leq p}$ where ${\displaystyle rb\leq p(m+rx)-f(m)}$ follows from ${\displaystyle b\leq c}$ when ${\displaystyle r>0}$ (respectively, follows from ${\displaystyle a\leq b}$ when ${\displaystyle r<0}$). ${\displaystyle \blacksquare }$

The lemma above is the key step in deducing the dominated extension theorem from Zorn's lemma.

Proof of dominated extension theorem using Zorn's lemma

The set of all possible dominated linear extensions of ${\displaystyle f}$ are partially ordered by extension of each other, so there is a maximal extension ${\displaystyle F.}$ By the codimension-1 result, if ${\displaystyle F}$ is not defined on all of ${\displaystyle X,}$ then it can be further extended. Thus ${\displaystyle F}$ must be defined everywhere, as claimed. ${\displaystyle \blacksquare }$

When ${\displaystyle M}$ 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[11] (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 Hausdorff spaces[12] (which is also equivalent to the ultrafilter lemma)

The Mizar project has completely formalized and automatically checked the proof of the Hahn–Banach theorem in the HAHNBAN file.[13]

The key element of the Hahn–Banach theorem is fundamentally a result about the separation of two convex sets: ${\displaystyle \{-p(-x-n)-f(n):n\in M\},}$ and ${\displaystyle \{p(m+x)-f(m):m\in M\}.}$ This sort of argument appears widely in convex geometry,[14] optimization theory, and economics. Lemmas to this end derived from the original Hahn–Banach theorem are known as the Hahn–Banach separation theorems.[15][16]

When the convex sets have additional properties, such as being open or compact for example, then the conclusion can be substantially strengthened:

Then following important corollary is known as the Geometric Hahn–Banach theorem or Mazur's theorem. It follows from the first bullet above and the convexity of ${\displaystyle M.}$

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

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

For example, linear subspaces are characterized by functionals: if X is a normed vector space with linear subspace M (not necessarily closed) and if ${\displaystyle z}$ is an element of X not in the closure of M, then there exists a continuous linear map ${\displaystyle f:X\to \mathbf {K} }$ with ${\displaystyle f(m)=0}$ for all ${\displaystyle m\in M,}$ ${\displaystyle f(z)=1,}$ and ${\displaystyle \|f\|=\operatorname {dist} (z,M)^{-1}.}$ (To see this, note that ${\displaystyle \operatorname {dist} (\cdot ,M)}$ is a sublinear function.) Moreover, if ${\displaystyle z}$ is an element of X, then there exists a continuous linear map ${\displaystyle f:X\to \mathbf {K} }$ such that ${\displaystyle f(z)=\|z\|}$ and ${\displaystyle \|f\|\leq 1.}$ This implies that the natural injection ${\displaystyle J}$ from a normed space X into its double dual ${\displaystyle V^{**}}$ 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 or locally convex. However, suppose X is a topological vector space, not necessarily Hausdorff or locally convex, but with a nonempty, proper, convex, open set M. Then geometric Hahn-Banach implies that there is a hyperplane separating M from any other point. In particular, there must exist a nonzero functional on X — that is, the continuous dual space ${\displaystyle X^{*}}$ is non-trivial.[3][20] Considering X with the weak topology induced by ${\displaystyle X^{*},}$ then X becomes locally convex; by the second bullet of geometric Hahn-Banach, the weak topology on this new space separates points. Thus X with this weak topology becomes Hausdorff. This sometimes allows some results from locally convex topological vector spaces to be applied to non-Hausdorff and non-locally convex spaces.

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:

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

Geometric separation

Hahn–Banach sandwich theorem[3] — Let ${\displaystyle p:X\to \mathbb {R} }$ be a sublinear function on a real vector space ${\displaystyle X,}$ let ${\displaystyle S\subseteq X}$ be any subset of ${\displaystyle X,}$ and let ${\displaystyle f:S\to \mathbb {R} }$ be any map. If there exist positive real numbers ${\displaystyle a}$ and ${\displaystyle b}$ such that

$${\displaystyle 0\geq \inf _{s\in S}[p(s-ax-by)-f(s)-af(x)-bf(y)]\qquad {\text{ for all }}x,y\in S,}$$

then there exists a linear functional ${\displaystyle F:X\to \mathbb {R} }$ on ${\displaystyle X}$ such that ${\displaystyle F\leq p}$ on ${\displaystyle X}$ and ${\displaystyle f\leq F\leq p}$ on ${\displaystyle S.}$