Infunctional analysis, theHahn–Banach theorem is a central result that allows the extension ofbounded linear functionals defined on avector subspace of somevector space to the whole space. The theorem also shows that there are sufficientcontinuous linear functionals defined on everynormed vector space in order to study thedual space. Another version of the Hahn–Banach theorem is known as theHahn–Banach separation theorem or thehyperplane separation theorem, and has numerous uses inconvex geometry.

The theorem is named for the mathematiciansHans Hahn andStefan 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) byEduard Helly,^[1] and a more general extension theorem, theM. Riesz extension theorem, from which the Hahn–Banach theorem can be derived, was proved in 1923 byMarcel Riesz.^[2]

The first Hahn–Banach theorem was proved byEduard Helly in 1912 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-dimensionalextension exists (where the linear functional has its domain extended by one dimension) and then usinginduction. In 1927, Hahn defined generalBanach 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 usessublinear functions. Whereas Helly's proof used mathematical induction, Hahn and Banach both usedtransfinite induction.^[3]

The Hahn–Banach theorem arose from attempts to solve infinite systems of linear equations. This is needed to solve problems such as themoment problem, whereby given all the potentialmoments 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 theFourier 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 anormed 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 areflexive 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 () 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]

The Hahn–Banach theorem can be deduced from the above theorem.^[3] If${\displaystyle X}$ isreflexive 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 bedominated (above) by a function${\displaystyle p:X\to \mathbb{R}}$ if${\displaystyle f(m)\le p(m)}$ for every${\displaystyle m\in M.}$ For this reason, the following version of the Hahn–Banach theorem is calledthe dominatedextension theorem.

The theorem remains true if the requirements on${\displaystyle p}$ are relaxed to require only that${\displaystyle p}$ be aconvex function:^[7][8] A function${\displaystyle p:X\to \mathbb{R}}$ is convex and satisfies${\displaystyle p(0)\le 0}$ if and only if${\displaystyle p(ax+by)\le ap(x)+bp(y)}$ for all vectors${\displaystyle x,y\in X}$ and all non-negative real${\displaystyle a,b\ge 0}$ such that${\displaystyle a+b\le 1.}$ Everysublinear function is a convex function. On the other hand, if${\displaystyle p:X\to \mathbb{R}}$ is convex with${\displaystyle p(0)\ge 0,}$ then the function defined by${\displaystyle p_0(x)\stackrel{\text{def}}{=}\underset{t>0}{inf}\frac{p(tx)}{t}}$ ispositively homogeneous (because for all${\displaystyle x}$ and${\displaystyle r>0}$ one has${\displaystyle p_0(rx)=\underset{t>0}{inf}\frac{p(trx)}{t}=r\underset{t>0}{inf}\frac{p(trx)}{tr}=r\underset{\tau >0}{inf}\frac{p(\tau x)}{\tau }=r{p}_0(x)}$), hence, being convex,it is sublinear. It is also bounded above by${\displaystyle p_0\le p,}$ and satisfies${\displaystyle F\le p_0}$ for every linear functional${\displaystyle F\le p.}$ 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${\displaystyle F:X\to \mathbb{R}}$ is linear then${\displaystyle F\le p}$ if and only if^[4]$${\displaystyle -p(-x)\le F(x)\le p(x)\text{ for all }x\in X,}$$ which is the (equivalent) conclusion that some authors^[4] write instead of${\displaystyle F\le p.}$It follows that if${\displaystyle p:X\to \mathbb{R}}$ is alsosymmetric, meaning that${\displaystyle p(-x)=p(x)}$ holds for all${\displaystyle x\in X,}$ then${\displaystyle F\le p}$ if and only${\displaystyle |F|\le p.}$ Everynorm is aseminorm and both are symmetricbalanced sublinear functions. A sublinear function is a seminorm if and only if it is abalanced 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. Theidentity function${\displaystyle \mathbb{R}\to \mathbb{R}}$ on${\displaystyle X:=\mathbb{R}}$ is an example of a sublinear function that is not a seminorm.

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.

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${\displaystyle |a|+|b|\le 1,}$^[8]$${\displaystyle p(ax+by)\le |a|p(x)+|b|p(y).}$$This condition holds if and only if${\displaystyle p}$ is aconvex andbalanced function satisfying${\displaystyle p(0)\le 0,}$ or equivalently, if and only if it is convex, satisfies${\displaystyle p(0)\le 0,}$ and${\displaystyle p(ux)\le p(x)}$ for all${\displaystyle x\in X}$ and allunit length scalars${\displaystyle u.}$

A complex-valued functional${\displaystyle F}$ is said to bedominated by${\displaystyle p}$ if${\displaystyle |F(x)|\le p(x)}$ 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 aseminorm defined on a real or complex vector space${\displaystyle X,}$ then everydominated 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 aconvex orsublinear function, this conclusion will remain true if both instances of "dominated" (meaning${\displaystyle |F|\le p}$) are weakened to instead mean "dominatedabove" (meaning${\displaystyle F\le p}$).^[7][8]

Proof

The following observations allow theHahn–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 iscompletely determined by itsreal part${\displaystyle \mathrm{Re}F:X\to \mathbb{R}}$ through the formula^{[6][proof 1]}$${\displaystyle F(x)=\mathrm{Re}F(x)-i\mathrm{Re}F(ix)\text{ for all }x\in X}$$and moreover, if${\displaystyle \Vert \cdot \Vert }$ is anorm on${\displaystyle X}$ then theirdual norms are equal:${\displaystyle \Vert F\Vert =\Vert \mathrm{Re}F\Vert .}$^[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${\displaystyle \mathrm{Re}F}$ extends${\displaystyle \mathrm{Re}f}$). The real part of a linear functional on${\displaystyle X}$ is always areal-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]}$${\displaystyle |F|\le p\text{ if and only if }\mathrm{Re}F\le p.}$$Stated in simpler language, a linear functional isdominated by a seminorm${\displaystyle p}$ if and only if itsreal part is dominated above by${\displaystyle p.}$

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${\displaystyle \mathrm{Re}f:M\to \mathbb{R};}$ 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 atopological vector space iscontinuous if and only if this is true of its real part${\displaystyle \mathrm{Re}F;}$ if the domain is a normed space then${\displaystyle \Vert F\Vert =\Vert \mathrm{Re}F\Vert }$ (where one side is infinite if and only if the other side is infinite).^[10] Assume${\displaystyle X}$ is atopological vector space and${\displaystyle p:X\to \mathbb{R}}$ issublinear function. If${\displaystyle p}$ is acontinuous 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 itsabsolute value${\displaystyle |F|}$ (which is aseminorm 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.

TheHahn–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${\displaystyle M}$ hascodimension${\displaystyle 1.}$^[3]

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

Thelemma above is the key step in deducing the dominated extension theorem fromZorn's lemma.

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 usesZorn's lemma although the strictly weakerultrafilter lemma^[11] (which is equivalent to thecompactness theorem and to theBoolean prime ideal theorem) may be used instead. Hahn–Banach can also be proved usingTychonoff's theorem forcompactHausdorff spaces^[12] (which is also equivalent to the ultrafilter lemma)

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

The Hahn–Banach theorem can be used to guarantee the existence ofcontinuous linear extensions ofcontinuous linear functionals.

Incategory-theoretic terms, the underlying field of the vector space is aninjective object in the category of locally convex vector spaces.

On anormed (orseminormed) space, a linear extension${\displaystyle F}$ of abounded linear functional${\displaystyle f}$ is said to benorm-preserving if it has the samedual norm as the original functional:${\displaystyle \Vert F\Vert =\Vert f\Vert .}$Because of this terminology, the second part ofthe above theorem is sometimes referred to as the "norm-preserving" version of the Hahn–Banach theorem.^[15] Explicitly:

The following observations allow thecontinuous extension theorem to be deduced from theHahn–Banach theorem.^[16]

The absolute value of a linear functional is always a seminorm. A linear functional${\displaystyle F}$ on atopological vector space${\displaystyle X}$ is continuous if and only if its absolute value${\displaystyle |F|}$ is continuous, which happens if and only if there exists a continuous seminorm${\displaystyle p}$ on${\displaystyle X}$ such that${\displaystyle |F|\le p}$ on the domain of${\displaystyle F.}$^[17] If${\displaystyle X}$ is a locally convex space then this statement remains true when the linear functional${\displaystyle F}$ is defined on aproper vector subspace of${\displaystyle X.}$

Proof for normed spaces

A linear functional${\displaystyle f}$ on anormed space iscontinuous if and only if it isbounded, which means that itsdual norm$${\displaystyle \Vert f\Vert =sup\{|f(m)|:\Vert m\Vert \le 1,m\in \mathrm{domain}f\}}$$is finite, in which case${\displaystyle |f(m)|\le \Vert f\Vert \Vert m\Vert }$ holds for every point${\displaystyle m}$ in its domain. Moreover, if${\displaystyle c\ge 0}$ is such that${\displaystyle |f(m)|\le c\Vert m\Vert }$ for all${\displaystyle m}$ in the functional's domain, then necessarily${\displaystyle \Vert f\Vert \le c.}$If${\displaystyle F}$ is a linear extension of a linear functional${\displaystyle f}$ then their dual norms always satisfy${\displaystyle \Vert f\Vert \le \Vert F\Vert }$^{[proof 3]} so that equality${\displaystyle \Vert f\Vert =\Vert F\Vert }$ is equivalent to${\displaystyle \Vert F\Vert \le \Vert f\Vert ,}$ which holds if and only if${\displaystyle |F(x)|\le \Vert f\Vert \Vert x\Vert }$ for every point${\displaystyle x}$ in the extension's domain. This can be restated in terms of the function${\displaystyle \Vert f\Vert \Vert \cdot \Vert :X\to \mathbb{R}}$ defined by${\displaystyle x\mapsto \Vert f\Vert \Vert x\Vert ,}$ which is always aseminorm:^{[note 4]}

A linear extension of abounded linear functional${\displaystyle f}$ isnorm-preserving if and only if the extension isdominated by the seminorm${\displaystyle \Vert f\Vert \Vert \cdot \Vert .}$

Applying theHahn–Banach theorem to${\displaystyle f}$ with this seminorm${\displaystyle \Vert f\Vert \Vert \cdot \Vert }$ thus produces a dominated linear extension whose norm is (necessarily) equal to that of${\displaystyle f,}$ which proves the theorem:

Thecontinuous extension theorem might fail if thetopological vector space (TVS)${\displaystyle X}$ is notlocally convex. For example, for theLebesgue space${\displaystyle L^p([0,1])}$ is acompletemetrizable TVS (anF-space) that isnot locally convex (in fact, its only convex open subsets are itself${\displaystyle L^p([0,1])}$ and the empty set) and the only continuous linear functional on${\displaystyle L^p([0,1])}$ is the constant${\displaystyle 0}$ function (Rudin 1991, §1.47). Since${\displaystyle L^p([0,1])}$ is Hausdorff, every finite-dimensional vector subspace${\displaystyle M\subseteq L^p([0,1])}$ islinearly homeomorphic toEuclidean space${\displaystyle {\mathbb{R}}^{\dim M}}$ or${\displaystyle {\mathbb{C}}^{\dim M}}$ (byF. Riesz's theorem) and so every non-zero linear functional${\displaystyle f}$ on${\displaystyle M}$ is continuous but none has a continuous linear extension to all of${\displaystyle L^p([0,1]).}$ However, it is possible for a TVS${\displaystyle X}$ to not be locally convex but nevertheless have enough continuous linear functionals that itscontinuous dual space${\displaystyle X^{\ast }}$separates points; for such a TVS, a continuous linear functional defined on a vector subspacemight have a continuous linear extension to the whole space.

If theTVS${\displaystyle X}$ is notlocally convex then there might not exist any continuous seminorm${\displaystyle p:X\to \mathbb{R}}$defined on${\displaystyle X}$ (not just on${\displaystyle M}$) that dominates${\displaystyle f,}$ in which case the Hahn–Banach theorem can not be applied as it was inthe 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${\displaystyle X}$ is any TVS (not necessarily locally convex), then a continuous linear functional${\displaystyle f}$ defined on a vector subspace${\displaystyle M}$ has a continuous linear extension${\displaystyle F}$ to all of${\displaystyle X}$ if and only if there exists some continuous seminorm${\displaystyle p}$ on${\displaystyle X}$ thatdominates${\displaystyle f.}$ Specifically, if given a continuous linear extension${\displaystyle F}$ then${\displaystyle p:=|F|}$ is a continuous seminorm on${\displaystyle X}$ that dominates${\displaystyle f;}$ and conversely, if given a continuous seminorm${\displaystyle p:X\to \mathbb{R}}$ on${\displaystyle X}$ that dominates${\displaystyle f}$ then any dominated linear extension of${\displaystyle f}$ to${\displaystyle X}$ (the existence of which is guaranteed by the Hahn–Banach theorem) will be a continuous linear extension.

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 inconvex geometry,^[18]optimization theory, andeconomics. Lemmas to this end derived from the original Hahn–Banach theorem are known as theHahn–Banach separation theorems.^[19][20] They are generalizations of thehyperplane separation theorem, which states that two disjoint nonempty convex subsets of a finite-dimensional space${\displaystyle {\mathbb{R}}^n}$ can be separated by someaffine hyperplane, which is afiber (level set) of the form${\displaystyle f^{-1}(s)=\{x:f(x)=s\}}$ where${\displaystyle f\ne 0}$ is a non-zero linear functional and${\displaystyle s}$ is a scalar.

When the convex sets have additional properties, such as beingopen orcompact for example, then the conclusion can be substantially strengthened:

Then following important corollary is known as theGeometric Hahn–Banach theorem orMazur's theorem (also known asAscoli–Mazur theorem^[22]). 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.

Since points are triviallyconvex, geometric Hahn–Banach implies that functionals can detect theboundary of a set. In particular, let${\displaystyle X}$ be a real topological vector space and${\displaystyle A\subseteq X}$ be convex with${\displaystyle \mathrm{Int}A\ne \varnothing .}$ If${\displaystyle a_0\in A\setminus \mathrm{Int}A}$ then there is a functional that is vanishing at${\displaystyle a_0,}$ but supported on the interior of${\displaystyle A.}$^[19]

Call a normed space${\displaystyle X}$smooth if at each point${\displaystyle x}$ in its unit ball there exists a unique closed hyperplane to the unit ball at${\displaystyle x.}$ Köthe showed in 1983 that a normed space is smooth at a point${\displaystyle x}$ if and only if the norm isGateaux differentiable at that point.^[3]

Let${\displaystyle U}$ be a convexbalanced neighborhood of the origin in alocally convex topological vector space${\displaystyle X}$ and suppose${\displaystyle x\in X}$ is not an element of${\displaystyle U.}$ Then there exists a continuous linear functional${\displaystyle f}$ on${\displaystyle X}$ such that^[3]${\displaystyle sup|f(U)|\le |f(x)|.}$

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

For example, linear subspaces are characterized by functionals: ifX is a normed vector space with linear subspaceM (not necessarily closed) and if${\displaystyle z}$ is an element ofX not in theclosure ofM, 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${\displaystyle \Vert f\Vert =\mathrm{dist}(z,M{)}^{-1}.}$ (To see this, note that${\displaystyle \mathrm{dist}(\cdot ,M)}$ is a sublinear function.) Moreover, if${\displaystyle z}$ is an element ofX, then there exists a continuous linear map${\displaystyle f:X\to \mathbf{K}}$ such that${\displaystyle f(z)=\Vert z\Vert }$ and${\displaystyle \Vert f\Vert \le 1.}$ This implies that thenatural injection${\displaystyle J}$ from a normed spaceX into itsdouble dual${\displaystyle V^{\ast \ast }}$ 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 isHausdorff orlocally convex. However, supposeX is a topological vector space, not necessarily Hausdorff orlocally convex, but with a nonempty, proper, convex, open setM. Then geometric Hahn–Banach implies that there is a hyperplane separatingM from any other point. In particular, there must exist a nonzero functional onX — that is, thecontinuous dual space${\displaystyle X^{\ast }}$ is non-trivial.^[3][25] ConsideringX with theweak topology induced by${\displaystyle X^{\ast },}$ thenX becomes locally convex; by the second bullet of geometric Hahn–Banach, the weak topology on this new spaceseparates points. ThusX with this weak topology becomesHausdorff. This sometimes allows some results from locally convex topological vector spaces to be applied to non-Hausdorff and non-locally convex spaces.

The Hahn–Banach theorem is often useful when one wishes to apply the method ofa priori estimates. Suppose that we wish to solve the linear differential equation${\displaystyle Pu=f}$ for${\displaystyle u,}$ with${\displaystyle f}$ given in some Banach spaceX. If we have control on the size of${\displaystyle u}$ in terms of${\displaystyle \Vert f{\Vert }_X}$ and we can think of${\displaystyle u}$ as a bounded linear functional on some suitable space of test functions${\displaystyle g,}$ then we can view${\displaystyle f}$ as a linear functional by adjunction:${\displaystyle (f,g)=(u,P^{\ast }g).}$ At first, this functional is only defined on the image of${\displaystyle P,}$ but using the Hahn–Banach theorem, we can try to extend it to the entire codomainX. The resulting functional is often defined to be aweak solution to the equation.