# 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 "abbastanza" 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.

Contenuti 1 Storia 2 Hahn–Banach theorem 2.1 For complex or real vector spaces 2.2 Prova 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 Applicazioni 4.1 Partial differential equations 4.2 Characterizing reflexive Banach spaces 4.3 Example from Fredholm theory 5 generalizzazioni 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 conversare 7 Relation to axiom of choice and other theorems 8 Guarda anche 9 Appunti 10 Riferimenti 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 {stile di visualizzazione C[un,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, e, 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, e, ancora, find it if so.

Riesz and Helly solved the problem for certain classes of spaces (come {stile di visualizzazione L^{p}([0,1])} e {stile di visualizzazione C([un,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 {stile di visualizzazione a sinistra(f_{io}Giusto)_{iin I}} of bounded linear functionals on a normed space {stile di visualizzazione X} and a collection of scalars {stile di visualizzazione a sinistra(c_{io}Giusto)_{iin I},} determine if there is an {stile di visualizzazione xin X} tale che {stile di visualizzazione f_{io}(X)=c_{io}} per tutti {displaystyle iin I.} Se {stile di visualizzazione 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 {stile di visualizzazione a sinistra(X_{io}Giusto)_{iin I}} of vectors in a normed space {stile di visualizzazione X} and a collection of scalars {stile di visualizzazione a sinistra(c_{io}Giusto)_{iin I},} determine if there is a bounded linear functional {stile di visualizzazione f} Su {stile di visualizzazione X} tale che {stile di visualizzazione deviato(X_{io}Giusto)=c_{io}} per tutti {displaystyle iin I.} Riesz went on to define {stile di visualizzazione L^{p}([0,1])} spazio ( {stile di visualizzazione 10} such that for any choice of scalars {stile di visualizzazione a sinistra(S_{io}Giusto)_{iin I}} where all but finitely many {stile di visualizzazione s_{io}} sono {stile di visualizzazione 0,} the following holds: {stile di visualizzazione a sinistra|somma _{iin I}S_{io}c_{io}Giusto|leq Kleft|somma _{iin I}S_{io}X_{io}Giusto|.} The Hahn–Banach theorem can be deduced from the above theorem.[3] Se {stile di visualizzazione X} is reflexive then this theorem solves the vector problem.

Hahn–Banach theorem A real-valued function {stile di visualizzazione f:fiume matematicabb {R} } defined on a subset {stile di visualizzazione M} di {stile di visualizzazione X} is said to be dominated (sopra) by a function {stile di visualizzazione p:Xto matematicabb {R} } Se {stile di visualizzazione f(m)leq p(m)} per ogni {displaystyle min M.} Hence the reason why the following version of the Hahn-Banach theorem is called the dominated extension theorem.

Per esempio, linear subspaces are characterized by functionals: if X is a normed vector space with linear subspace M (not necessarily closed) e se {stile di visualizzazione con} is an element of X not in the closure of M, then there exists a continuous linear map {stile di visualizzazione f:Xto mathbf {K} } insieme a {stile di visualizzazione f(m)=0} per tutti {displaystyle min M,} {stile di visualizzazione f(z)=1,} e {stile di visualizzazione |f|=nome operatore {dist} (z,M)^{-1}.} (Per vedere questo, note that {nome dell'operatore dello stile di visualizzazione {dist} (cdot ,M)} is a sublinear function.) Inoltre, Se {stile di visualizzazione con} is an element of X, then there exists a continuous linear map {stile di visualizzazione f:Xto mathbf {K} } tale che {stile di visualizzazione f(z)=|z|} e {stile di visualizzazione |f|leq 1.} This implies that the natural injection {stile di visualizzazione J} from a normed space X into its double dual {stile di visualizzazione 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. Per esempio, many results in functional analysis assume that a space is Hausdorff or locally convex. Tuttavia, suppose X is a topological vector space, non necessariamente Hausdorff o localmente convesso, but with a nonempty, corretto, convex, open set M. Then geometric Hahn-Banach implies that there is a hyperplane separating M from any other point. In particolare, there must exist a nonzero functional on X — that is, the continuous dual space {stile di visualizzazione X^{*}} is non-trivial.[3][20] Considering X with the weak topology induced by {stile di visualizzazione 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} per {stile di visualizzazione u,} insieme a {stile di visualizzazione f} given in some Banach space X. If we have control on the size of {stile di visualizzazione u} in termini di {stile di visualizzazione |F|_{X}} and we can think of {stile di visualizzazione u} as a bounded linear functional on some suitable space of test functions {stile di visualizzazione g,} then we can view {stile di visualizzazione f} as a linear functional by adjunction: {stile di visualizzazione (f,g)=(tu,P^{*}g).} At first, this functional is only defined on the image of {stile di visualizzazione 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 {stile di visualizzazione X} is a Hausdorff locally convex TVS over the field {displaystyle mathbf {K} } e {stile di visualizzazione Y} is a vector subspace of {stile di visualizzazione X} that is TVS–isomorphic to {displaystyle mathbf {K} ^{io}} for some set {displaystyle I.} Quindi {stile di visualizzazione Y} is a closed and complemented vector subspace of {stile di visualizzazione X.} Proof Since {displaystyle mathbf {K} ^{io}} is a complete TVS so is {stile di visualizzazione Y,} and since any complete subset of a Hausdorff TVS is closed, {stile di visualizzazione Y} è un sottoinsieme chiuso di {stile di visualizzazione X.} Permettere {displaystyle f=left(f_{io}Giusto)_{iin I}:Yto mathbf {K} ^{io}} be a TVS isomorphism, so that each {stile di visualizzazione f_{io}:Yto mathbf {K} } is a continuous surjective linear functional. By the Hahn–Banach theorem, we may extend each {stile di visualizzazione f_{io}} to a continuous linear functional {stile di visualizzazione F_{io}:Xto mathbf {K} } Su {stile di visualizzazione X.} Permettere {stile di visualizzazione F:= sinistra(F_{io}Giusto)_{iin I}:Xto mathbf {K} ^{io}} Così {stile di visualizzazione F} is a continuous linear surjection such that its restriction to {stile di visualizzazione Y} è {stile di visualizzazione F{big vert }_{Y}= sinistra(F_{io}{big vert }_{Y}Giusto)_{iin I}= sinistra(f_{io}Giusto)_{iin I}=f.} Permettere {stile di visualizzazione P:=f^{-1}circ F:Xth Y,} which is a continuous linear map whose restriction to {stile di visualizzazione Y} è {stile di visualizzazione P{big vert }_{Y}=f^{-1}circ F{big vert }_{Y}=f^{-1}circ f=mathbf {1} _{Y},} dove {displaystyle mathbb {1} _{Y}} denotes the identity map on {stile di visualizzazione Y.} This shows that {stile di visualizzazione P} is a continuous linear projection onto {stile di visualizzazione Y} (questo è, {displaystyle Pcirc P=P} ). così {stile di visualizzazione Y} is complemented in {stile di visualizzazione X} e {displaystyle X=Yoplus ker P} in the category of TVSs. {stile di visualizzazione quadrato nero } 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: {stile di visualizzazione p:Xto matematicabb {R} } is a sublinear function (possibly a seminorm) on a vector space {stile di visualizzazione X,} {stile di visualizzazione M} is a vector subspace of {stile di visualizzazione X} (possibly closed), e {stile di visualizzazione f} is a linear functional on {stile di visualizzazione M} soddisfacente {stile di visualizzazione |f|leq p} Su {stile di visualizzazione M} (and possibly some other conditions). One then concludes that there exists a linear extension {stile di visualizzazione F} di {stile di visualizzazione f} a {stile di visualizzazione X} tale che {stile di visualizzazione |F|leq p} Su {stile di visualizzazione X} (possibly with additional properties).

Teorema[3] — If {stile di visualizzazione D} is an absorbing disk in a real or complex vector space {stile di visualizzazione X} e se {stile di visualizzazione f} be a linear functional defined on a vector subspace {stile di visualizzazione M} di {stile di visualizzazione X} tale che {stile di visualizzazione |f|leq 1} Su {displaystyle Mcap D,} then there exists a linear functional {stile di visualizzazione F} Su {stile di visualizzazione X} extending {stile di visualizzazione f} tale che {stile di visualizzazione |F|leq 1} Su {displaystyle D.} For seminorms Hahn–Banach theorem for seminorms[22][23] — If {stile di visualizzazione p:fiume matematicabb {R} } is a seminorm defined on a vector subspace {stile di visualizzazione M} di {stile di visualizzazione X,} e se {stile di visualizzazione q:Xto matematicabb {R} } is a seminorm on {stile di visualizzazione X} tale che {displaystyle pleq q{big vert }_{M},} then there exists a seminorm {stile di visualizzazione P:Xto matematicabb {R} } Su {stile di visualizzazione X} tale che {stile di visualizzazione P{big vert }_{M}= p} Su {stile di visualizzazione M} e {displaystyle Pleq q} Su {stile di visualizzazione X.} Proof of the Hahn–Banach theorem for seminorms Let {stile di visualizzazione S} be the convex hull of {stile di visualizzazione {min m:p(m)leq 1}tazza {xin X:q(X)leq 1}.} Perché {stile di visualizzazione S} is an absorbing disk in {stile di visualizzazione X,} its Minkowski functional {stile di visualizzazione P} is a seminorm. Quindi {displaystyle p=P} Su {stile di visualizzazione M} e {displaystyle Pleq q} Su {stile di visualizzazione X.} Geometric separation Hahn–Banach sandwich theorem[3] - Permettere {stile di visualizzazione p:Xto matematicabb {R} } be a sublinear function on a real vector space {stile di visualizzazione X,} permettere {displaystyle Ssubseteq X} be any subset of {stile di visualizzazione X,} e lascia {stile di visualizzazione f:Sto mathbb {R} } be any map. If there exist positive real numbers {stile di visualizzazione a} e {stile di visualizzazione b} tale che {displaystyle 0geq inf _{peccato S}[p(s-ax-by)-f(S)-af(X)-bf(y)]qquad {testo{ per tutti }}X,yin S,} then there exists a linear functional {stile di visualizzazione F:Xto matematicabb {R} } Su {stile di visualizzazione X} tale che {displaystyle Fleq p} Su {stile di visualizzazione X} e {displaystyle fleq Fleq p} Su {displaystyle S.} Maximal dominated linear extension Theorem[3] (Andenaes, 1970) - Permettere {stile di visualizzazione p:Xto matematicabb {R} } be a sublinear function on a real vector space {stile di visualizzazione X,} permettere {stile di visualizzazione f:fiume matematicabb {R} } be a linear functional on a vector subspace {stile di visualizzazione M} di {stile di visualizzazione X} tale che {displaystyle fleq p} Su {stile di visualizzazione M,} e lascia {displaystyle Ssubseteq X} be any subset of {stile di visualizzazione X.} Then there exists a linear functional {stile di visualizzazione F:Xto matematicabb {R} } Su {stile di visualizzazione X} that extends {stile di visualizzazione f,} soddisfa {displaystyle Fleq p} Su {stile di visualizzazione X,} and is (pointwise) maximal on {stile di visualizzazione S} nel seguente senso: Se {stile di visualizzazione {widehat {F}}:Xto matematicabb {R} } is a linear functional on {stile di visualizzazione X} that extends {stile di visualizzazione f} e soddisfa {stile di visualizzazione {widehat {F}}leq p} Su {stile di visualizzazione X,} poi {displaystyle Fleq {widehat {F}}} Su {stile di visualizzazione S} implica {displaystyle F={widehat {F}}} Su {displaystyle S.} Se {stile di visualizzazione S={S}} is a singleton set (dove {displaystyle sin X} is some vector) e se {stile di visualizzazione F:Xto matematicabb {R} } is such a maximal dominated linear extension of {stile di visualizzazione f:fiume matematicabb {R} ,} poi {stile di visualizzazione 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 {stile di visualizzazione X} e {stile di visualizzazione Y} are vector spaces over the same field and if {stile di visualizzazione f:Mto Y} be a linear map defined on a vector subspace {stile di visualizzazione M} di {stile di visualizzazione X,} then there exists a linear map {stile di visualizzazione F:Xth Y} that extends {stile di visualizzazione f.} For nonlinear functions The following theorem of Mazur–Orlicz (1953) is equivalent to the Hahn–Banach theorem.

Mazur–Orlicz theorem[3] - Permettere {stile di visualizzazione p:Xto matematicabb {R} } be a sublinear function on a real or complex vector space {stile di visualizzazione X,} permettere {stile di visualizzazione T} be any set, e lascia {stile di visualizzazione R:Tto mathbb {R} } e {stile di visualizzazione v:Tto X} be any maps. The following statements are equivalent: there exists a real-valued linear functional {stile di visualizzazione F} Su {stile di visualizzazione X} tale che {displaystyle Fleq p} Su {stile di visualizzazione X} e {displaystyle Rleq Fcirc v} Su {stile di visualizzazione T} ; for any finite sequence {stile di visualizzazione s_{1},ldot ,S_{n}} di {displaystyle n>0} non-negative real numbers, and any sequence {stile di visualizzazione t_{1},ldot ,t_{n}in T} di elementi di {stile di visualizzazione T,} {somma dello stile di visualizzazione _{io=1}^{n}S_{io}Rleft(t_{io}Giusto)leq pleft(somma _{io=1}^{n}S_{io}vleft(t_{io}Giusto)Giusto).} The following theorem characterizes when any scalar function on {stile di visualizzazione X} (not necessarily linear) has a continuous linear extension to all of {stile di visualizzazione X.} Teorema (The extension principle[24]) - Permettere {stile di visualizzazione f} a scalar-valued function on a subset {stile di visualizzazione S} of a topological vector space {stile di visualizzazione X.} Then there exists a continuous linear functional {stile di visualizzazione F} Su {stile di visualizzazione X} extending {stile di visualizzazione f} if and only if there exists a continuous seminorm {stile di visualizzazione p} Su {stile di visualizzazione X} tale che {stile di visualizzazione a sinistra|somma _{io=1}^{n}un_{io}f(S_{io})Giusto|leq pleft(somma _{io=1}^{n}un_{io}S_{io}Giusto)} for all positive integers {stile di visualizzazione n} and all finite sequences {stile di visualizzazione a_{1},ldot ,un_{n}} of scalars and elements {stile di visualizzazione s_{1},ldot ,S_{n}} di {displaystyle S.} Converse Let X be a topological vector space. A vector subspace M of X has the extension property if any continuous linear functional on M can be extended to a continuous linear functional on X, and we say that X has the Hahn–Banach extension property (HBEP) if every vector subspace of X has the extension property.[25] The Hahn–Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable topological vector spaces there is a converse, due to Kalton: every complete metrizable TVS with the Hahn–Banach extension property is locally convex.[25] D'altro canto, a vector space 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.[25] A vector subspace M of a TVS X has the separation property if for every element of X such that {displaystyle xnot in M,} there exists a continuous linear functional {stile di visualizzazione f} on X such that {stile di visualizzazione f(X)neq 0} e {stile di visualizzazione f(m)=0} per tutti {displaystyle min M.} Chiaramente, the continuous dual space of a TVS X separates points on X if and only if {stile di visualizzazione {0},} has the separation property. In 1992, Kakol proved that any infinite dimensional vector space X, there exist TVS-topologies on X that do not have the HBEP despite having enough continuous linear functionals for the continuous dual space to separate points on X. Tuttavia, if X is a TVS then every vector subspace of X has the extension property if and only if every vector subspace of X has the separation property.[25] Relation to axiom of choice and other theorems See also: Krein–Milman theorem § Relation to other statements The proof of the 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 (corrente alternata). It was discovered by Łoś and Ryll-Nardzewski[12] and independently by Luxemburg[11] 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.[26] The ultrafilter lemma is equivalent (under ZF) to the Banach–Alaoglu theorem,[27] which is another foundational theorem in functional analysis. Although the Banach–Alaoglu theorem implies HB,[28] it is not equivalent to it (said differently, the Banach–Alaoglu theorem is strictly stronger than HB). Tuttavia, HB is equivalent to a certain weakened version of the Banach–Alaoglu theorem for normed spaces.[29] The Hahn–Banach theorem is also equivalent to the following statement:[30] (): On every Boolean algebra B there exists a "probability charge", questo è: a non-constant finitely additive map from {stile di visualizzazione B} in {stile di visualizzazione [0,1].} (BPI is equivalent to the statement that there are always non-constant probability charges which take only the values 0 e 1.) In ZF, the Hahn–Banach theorem suffices to derive the existence of a non-Lebesgue measurable set.[31] Inoltre, the Hahn–Banach theorem implies the Banach–Tarski paradox.[32] For separable Banach spaces, D. K. Brown and S. G. Simpson proved that the Hahn–Banach theorem follows from WKL0, a weak subsystem of second-order arithmetic that takes a form of Kőnig's lemma restricted to binary trees as an axiom. Infatti, they prove that under a weak set of assumptions, the two are equivalent, an example of reverse mathematics.[33][34] See also Farkas' lemma Fichera's existence principle – Theorem in functional analysis M. Riesz extension theorem Separating axis theorem Vector-valued Hahn–Banach theorems Notes ^ This definition means, per esempio, Quello {stile di visualizzazione F_{b}(X)=F_{b}(0+1X)=f(0)+1b=b} e se {displaystyle min M} poi {stile di visualizzazione F_{b}(m)=F_{b}(m+0x)=f(m)+0b=f(m).} Infatti, Se {stile di visualizzazione G:Moplus mathbb {R} xto mathbb {R} } is any linear extension of {stile di visualizzazione f} a {displaystyle Moplus mathbb {R} X} poi {displaystyle G=F_{b}} per {stile di visualizzazione b:=G(X).} In altre parole, every linear extension of {stile di visualizzazione f} a {displaystyle Moplus mathbb {R} X} is of the form {stile di visualizzazione F_{b}} per alcuni (unico) {displaystyle b.} ^ Explicitly, for any real number {displaystyle bin mathbb {R} ,} {stile di visualizzazione F_{b}leq p} Su {displaystyle Moplus mathbb {R} X} se e solo se {displaystyle aleq bleq c.} Combined with the fact that {stile di visualizzazione F_{b}(X)= b,} it follows that the dominated linear extension of {stile di visualizzazione f} a {displaystyle Moplus mathbb {R} X} is unique if and only if {displaystyle a=c,} in which case this scalar will be the extension's values at {displaystyle x.} Since every linear extension of {stile di visualizzazione f} a {displaystyle Moplus mathbb {R} X} is of the form {stile di visualizzazione F_{b}} per alcuni {stile di visualizzazione b,} the bounds {displaystyle aleq b=F_{b}(X)leq c} thus also limit the range of possible values (a {stile di visualizzazione x} ) that can be taken by any of {stile di visualizzazione f} 's dominated linear extensions. In particolare, Se {stile di visualizzazione F:Xto matematicabb {R} } is any linear extension of {stile di visualizzazione f} soddisfacente {displaystyle Fleq p} then for every {displaystyle xin Xsetminus M,} {stile di visualizzazione sup_{min m}[-p(-m-x)-f(m)]~leq ~F(X)~leq ~inf _{min m}[p(m+x)-f(m)].} ^ Geometric illustration: The geometric idea of the above proof can be fully presented in the case of {displaystyle X=matematicabb {R} ^{2},M={(X,0):xin mathbb {R} }.} Primo, define the simple-minded extension {stile di visualizzazione f_{0}(X,y)=f(X),} It doesn't work, since maybe {stile di visualizzazione f_{0}leq p} . But it is a step in the right direction. {displaystyle p-f_{0}} is still convex, e {displaystyle p-f_{0}geq f-f_{0}.} Ulteriore, {displaystyle f-f_{0}} is identically zero on the x-axis. Thus we have reduced to the case of {displaystyle f=0,pgeq 0} on the x-axis. Se {displaystyle pgeq 0} Su {displaystyle mathbb {R} ^{2},} allora abbiamo finito. Altrimenti, pick some {displaystyle vin mathbb {R} ^{2},} tale che {stile di visualizzazione p(v)<0.} The idea now is to perform a simultaneous bounding of {displaystyle p} on {displaystyle v+M} and {displaystyle -v+M} such that {displaystyle pgeq b} on {displaystyle v+M} and {displaystyle pgeq -b} on {displaystyle -v+M,} then defining {displaystyle {tilde {f}}(w+rv)=rb} would give the desired extension. Since {displaystyle -v+M,v+M} are on opposite sides of {displaystyle M,} and {displaystyle p<0} at some point on {displaystyle v+M,} by convexity of {displaystyle p,} we must have {displaystyle pgeq 0} on all points on {displaystyle -v+M.} Thus {displaystyle inf _{uin -v+M}p(u)} is finite. Geometrically, this works because {displaystyle {z:p(z)<0}} is a convex set that is disjoint from {displaystyle M,} and thus must lie entirely on one side of {displaystyle M.} Define {displaystyle b=-inf _{uin -v+M}p(u).} This satisfies {displaystyle pgeq -b} on {displaystyle -v+M.} It remains to check the other side. For all {displaystyle v+win v+M,} convexity implies that for all {displaystyle -v+w'in -v+M,p(v+w)+p(-v+w')geq 2p((w+w')/2)=0,} thus {displaystyle p(v+w)geq sup _{uin -v+M}-p(u)=b.} Since during the proof, we only used convexity of {displaystyle p} , we see that the lemma remains true for merely convex {displaystyle p.} Proofs ^ If {displaystyle z=a+ibin mathbb {C} } has real part {displaystyle operatorname {Re} z=a} then {displaystyle -ioperatorname {Re} (iz)=b,} which proves that {displaystyle z=operatorname {Re} z-ioperatorname {Re} (iz).} Substituting {displaystyle F(x)} in for {displaystyle z} and using {displaystyle iF(x)=F(ix)} gives {displaystyle F(x)=operatorname {Re} F(x)-ioperatorname {Re} F(ix).} {displaystyle blacksquare } ^ Let {displaystyle F} be any homogeneous scalar-valued map on {displaystyle X} (such as a linear functional) and let {displaystyle p:Xto mathbb {R} } be any map that satisfies {displaystyle p(ux)=p(x)} for all {displaystyle x} and unit length scalars {displaystyle u} (such as a seminorm). If {displaystyle |F|leq p} then {displaystyle operatorname {Re} Fleq |operatorname {Re} F|leq |F|leq p.} For the converse, assume {displaystyle operatorname {Re} Fleq p} and fix {displaystyle xin X.} Let {displaystyle r=|F(x)|} and pick any {displaystyle theta in mathbb {R} } such that {displaystyle F(x)=re^{itheta };} it remains to show {displaystyle rleq p(x).} Homogeneity of {displaystyle F} implies {displaystyle Fleft(e^{-itheta }xright)=r} is real so that {displaystyle operatorname {Re} Fleft(e^{-itheta }xright)=Fleft(e^{-itheta }xright).} By assumption, {displaystyle operatorname {Re} Fleq p} and {displaystyle pleft(e^{-itheta }xright)=p(x),} so that {displaystyle r=operatorname {Re} Fleft(e^{-itheta }xright)leq pleft(e^{-itheta }xright)=p(x),} as desired. {displaystyle blacksquare } References ^ O'Connor, John J.; Robertson, Edmund F., "Hahn–Banach theorem", MacTutor History of Mathematics archive, University of St Andrews ^ See M. Riesz extension theorem. According to Gȧrding, L. (1970). "Marcel Riesz in memoriam". Acta Math. 124 (1): I–XI. doi:10.1007/bf02394565. MR 0256837., the argument was known to Riesz already in 1918. ^ Jump up to: a b c d e f g h i j k l m n o p q r s t Narici & Beckenstein 2011, pp. 177–220. ^ Jump up to: a b c Rudin 1991, pp. 56–62. ^ Rudin 1991, Th. 3.2 ^ Jump up to: a b c d e f g h Narici & Beckenstein 2011, pp. 177–183. ^ Jump up to: a b c Schechter 1996, pp. 318–319. ^ Jump up to: a b c d Reed & Simon 1980. ^ Rudin 1991, Th. 3.2 ^ Jump up to: a b Narici & Beckenstein 2011, pp. 126–128. ^ Jump up to: a b Luxemburg 1962. ^ Jump up to: a b Łoś & Ryll-Nardzewski 1951, pp. 233–237. ^ HAHNBAN file ^ Harvey, R.; Lawson, H. B. (1983). "An intrinsic characterisation of Kähler manifolds". Invent. Math. 74 (2): 169–198. Bibcode:1983InMat..74..169H. doi:10.1007/BF01394312. S2CID 124399104. ^ Jump up to: a b c Zălinescu, C. (2002). Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing Co., Inc. pp. 5–7. ISBN 981-238-067-1. MR 1921556. ^ Gabriel Nagy, Real Analysis lecture notes ^ Brezis, Haim (2011). Functional Analysis, Sobolev Spaces, and Partial Differential Equations. New York: Springer. pp. 6–7. ^ Trèves 2006, p. 184. ^ Narici & Beckenstein 2011, pp. 195. ^ Schaefer & Wolff 1999, p. 47. ^ Narici & Beckenstein 2011, p. 212. ^ Wilansky 2013, pp. 18–21. ^ Narici & Beckenstein 2011, pp. 150. ^ Edwards 1995, pp. 124–125. ^ Jump up to: a b c d Narici & Beckenstein 2011, pp. 225–273. ^ Pincus 1974, pp. 203–205. ^ Schechter 1996, pp. 766–767. ^ Muger, Michael (2020). Topology for the Working Mathematician. ^ Bell, J.; Fremlin, David (1972). "A Geometric Form of the Axiom of Choice" (PDF). Fundamenta Mathematicae. 77 (2): 167–170. doi:10.4064/fm-77-2-167-170. Retrieved 26 Dec 2021. ^ Schechter, Eric. Handbook of Analysis and its Foundations. p. 620. ^ Foreman, M.; Wehrung, F. (1991). "The Hahn–Banach theorem implies the existence of a non-Lebesgue measurable set" (PDF). Fundamenta Mathematicae. 138: 13–19. doi:10.4064/fm-138-1-13-19. ^ Pawlikowski, Janusz (1991). "The Hahn–Banach theorem implies the Banach–Tarski paradox". Fundamenta Mathematicae. 138: 21–22. doi:10.4064/fm-138-1-21-22. ^ Brown, D. K.; Simpson, S. G. (1986). "Which set existence axioms are needed to prove the separable Hahn–Banach theorem?". Annals of Pure and Applied Logic. 31: 123–144. doi:10.1016/0168-0072(86)90066-7. Source of citation. ^ Simpson, Stephen G. (2009), Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, ISBN 978-0-521-88439-6, MR2517689 Bibliography Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. Banach, Stefan (1932). 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