# Hahn–Banach theorem

Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" 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.

Contents 1 History 2 Hahn–Banach theorem 2.1 For complex or real vector spaces 2.2 Proof 2.3 In locally convex spaces 3 Geometric Hahn–Banach (the Hahn–Banach separation theorems) 3.1 Supporting hyperplanes 3.2 Balanced or disked neighborhoods 4 Applications 4.1 Partial differential equations 4.2 Characterizing reflexive Banach spaces 4.3 Example from Fredholm theory 5 Generalizations 5.1 For seminorms 5.2 Geometric separation 5.3 Maximal dominated linear extension 5.4 Vector valued Hahn–Banach 5.5 For nonlinear functions 6 Converse 7 Relation to axiom of choice and other theorems 8 See also 9 Notes 10 References 11 Bibliography History 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)_{iin I}} of bounded linear functionals on a normed space {displaystyle X} and a collection of scalars {displaystyle left(c_{i}right)_{iin I},} determine if there is an {displaystyle xin X} such that {displaystyle f_{i}(x)=c_{i}} for all {displaystyle iin 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)_{iin I}} of vectors in a normed space {displaystyle X} and a collection of scalars {displaystyle left(c_{i}right)_{iin I},} determine if there is a bounded linear functional {displaystyle f} on {displaystyle X} such that {displaystyle fleft(x_{i}right)=c_{i}} for all {displaystyle iin I.} Riesz went on to define {displaystyle L^{p}([0,1])} space ( {displaystyle 10} such that for any choice of scalars {displaystyle left(s_{i}right)_{iin I}} where all but finitely many {displaystyle s_{i}} are {displaystyle 0,} the following holds: {displaystyle left|sum _{iin I}s_{i}c_{i}right|leq Kleft|sum _{iin 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.

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

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:Xto mathbf {K} } with {displaystyle f(m)=0} for all {displaystyle min 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:Xto 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.

Partial differential equations The Hahn–Banach theorem is often useful when one wishes to apply the method of a 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 space X. If we have control on the size of {displaystyle u} in terms of {displaystyle |F|_{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^{*}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 codomain X. The resulting functional is often defined to be a weak solution to the equation.

Characterizing reflexive Banach spaces Theorem[21] — A real Banach space is 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.

Proposition — Suppose {displaystyle X} is a Hausdorff locally convex TVS over the field {displaystyle mathbf {K} } and {displaystyle Y} is a vector subspace of {displaystyle X} that is TVS–isomorphic to {displaystyle mathbf {K} ^{I}} for some set {displaystyle I.} Then {displaystyle Y} is a closed and complemented vector subspace of {displaystyle X.} Proof Since {displaystyle mathbf {K} ^{I}} is a complete TVS so is {displaystyle Y,} and since any complete subset of a Hausdorff TVS is closed, {displaystyle Y} is a closed subset of {displaystyle X.} Let {displaystyle f=left(f_{i}right)_{iin I}:Yto mathbf {K} ^{I}} be a TVS isomorphism, so that each {displaystyle f_{i}:Yto mathbf {K} } is a continuous surjective linear functional. By the Hahn–Banach theorem, we may extend each {displaystyle f_{i}} to a continuous linear functional {displaystyle F_{i}:Xto mathbf {K} } on {displaystyle X.} Let {displaystyle F:=left(F_{i}right)_{iin I}:Xto mathbf {K} ^{I}} so {displaystyle F} is a continuous linear surjection such that its restriction to {displaystyle Y} is {displaystyle F{big vert }_{Y}=left(F_{i}{big vert }_{Y}right)_{iin I}=left(f_{i}right)_{iin I}=f.} Let {displaystyle P:=f^{-1}circ F:Xto Y,} which is a continuous linear map whose restriction to {displaystyle Y} is {displaystyle P{big vert }_{Y}=f^{-1}circ F{big vert }_{Y}=f^{-1}circ f=mathbf {1} _{Y},} where {displaystyle mathbb {1} _{Y}} denotes the identity map on {displaystyle Y.} This shows that {displaystyle P} is a continuous linear projection onto {displaystyle Y} (that is, {displaystyle Pcirc P=P} ). Thus {displaystyle Y} is complemented in {displaystyle X} and {displaystyle X=Yoplus ker P} in the category of TVSs. {displaystyle blacksquare } The above result may be used to show that every closed vector subspace of {displaystyle mathbb {R} ^{mathbb {N} }} is complemented because any such space is either finite dimensional or else TVS–isomorphic to {displaystyle mathbb {R} ^{mathbb {N} }.} 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: {displaystyle p:Xto 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).

Theorem[3] — If {displaystyle D} is an absorbing disk in a real or complex vector space {displaystyle X} and if {displaystyle f} be a linear functional defined on a vector subspace {displaystyle M} of {displaystyle X} such that {displaystyle |f|leq 1} on {displaystyle Mcap D,} then there exists a linear functional {displaystyle F} on {displaystyle X} extending {displaystyle f} such that {displaystyle |F|leq 1} on {displaystyle D.} For seminorms Hahn–Banach theorem for seminorms[22][23] — If {displaystyle p:Mto mathbb {R} } is a seminorm defined on a vector subspace {displaystyle M} of {displaystyle X,} and if {displaystyle q:Xto mathbb {R} } is a seminorm on {displaystyle X} such that {displaystyle pleq q{big vert }_{M},} then there exists a seminorm {displaystyle P:Xto mathbb {R} } on {displaystyle X} such that {displaystyle P{big vert }_{M}=p} on {displaystyle M} and {displaystyle Pleq q} on {displaystyle X.} Proof of the Hahn–Banach theorem for seminorms Let {displaystyle S} be the convex hull of {displaystyle {min M:p(m)leq 1}cup {xin X:q(x)leq 1}.} Because {displaystyle S} is an absorbing disk in {displaystyle X,} its Minkowski functional {displaystyle P} is a seminorm. Then {displaystyle p=P} on {displaystyle M} and {displaystyle Pleq q} on {displaystyle X.} Geometric separation Hahn–Banach sandwich theorem[3] — Let {displaystyle p:Xto mathbb {R} } be a sublinear function on a real vector space {displaystyle X,} let {displaystyle Ssubseteq X} be any subset of {displaystyle X,} and let {displaystyle f:Sto mathbb {R} } be any map. If there exist positive real numbers {displaystyle a} and {displaystyle b} such that {displaystyle 0geq inf _{sin S}[p(s-ax-by)-f(s)-af(x)-bf(y)]qquad {text{ for all }}x,yin S,} then there exists a linear functional {displaystyle F:Xto mathbb {R} } on {displaystyle X} such that {displaystyle Fleq p} on {displaystyle X} and {displaystyle fleq Fleq p} on {displaystyle S.} Maximal dominated linear extension Theorem[3] (Andenaes, 1970) — Let {displaystyle p:Xto mathbb {R} } be a sublinear function on a real vector space {displaystyle X,} let {displaystyle f:Mto mathbb {R} } be a linear functional on a vector subspace {displaystyle M} of {displaystyle X} such that {displaystyle fleq p} on {displaystyle M,} and let {displaystyle Ssubseteq X} be any subset of {displaystyle X.} Then there exists a linear functional {displaystyle F:Xto mathbb {R} } on {displaystyle X} that extends {displaystyle f,} satisfies {displaystyle Fleq p} on {displaystyle X,} and is (pointwise) maximal on {displaystyle S} in the following sense: if {displaystyle {widehat {F}}:Xto mathbb {R} } is a linear functional on {displaystyle X} that extends {displaystyle f} and satisfies {displaystyle {widehat {F}}leq p} on {displaystyle X,} then {displaystyle Fleq {widehat {F}}} on {displaystyle S} implies {displaystyle F={widehat {F}}} on {displaystyle S.} If {displaystyle S={s}} is a singleton set (where {displaystyle sin X} is some vector) and if {displaystyle F:Xto mathbb {R} } is such a maximal dominated linear extension of {displaystyle f:Mto mathbb {R} ,} then {displaystyle F(s)=inf _{min M}[f(s)+p(s-m)].} [3] Vector valued Hahn–Banach See also: Vector-valued Hahn–Banach theorems Vector–valued Hahn–Banach theorem[3] — If {displaystyle X} and {displaystyle Y} are vector spaces over the same field and if {displaystyle f:Mto Y} be a linear map defined on a vector subspace {displaystyle M} of {displaystyle X,} then there exists a linear map {displaystyle F:Xto Y} that extends {displaystyle f.} For nonlinear functions The following theorem of Mazur–Orlicz (1953) is equivalent to the Hahn–Banach theorem.