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.

Hahn–Banach dominated extension theorem (for real linear functionals)[4][5][6] — If {displaystyle p:Xto 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:Xto 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,yin X;{text{ and all real }};tgeq 0,} and if {displaystyle f:Mto 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 }}min M} then there exists a linear functional {displaystyle F:Xto mathbb {R} } such that {displaystyle F(m)=f(m)quad {text{ for all }}min M,} {displaystyle F(x)leq p(x)quad ~;,{text{ for all }}xin X.} Moreover, if {displaystyle p} is a seminorm then {displaystyle |F(x)|leq p(x)} necessarily holds for all {displaystyle xin 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 }}00} (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 rbleq p(m+rx)-f(m).} {displaystyle blacksquare } This lemma remains true if {displaystyle p:Xto mathbb {R} } is merely a convex function instead of a sublinear function.[7][8] show Proof 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 theoremfor 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] In locally convex spaces In the above form, the functional to be extended must already be bounded by a sublinear function. In some applications, this might close to begging the question. However, in locally convex spaces, any continuous functional is already bounded by the norm, which is sublinear. One thus has Continuous extensions on locally convex spaces[3] — Let X be locally convex topological vector space over {displaystyle mathbf {K} } (either {displaystyle mathbb {R} } or {displaystyle mathbb {C} } ), {displaystyle M} a vector subspace of X, and {displaystyle f} a continuous linear functional on {displaystyle M.} Then {displaystyle f} has a continuous linear extension to all of X. If the topology on X arises from a norm, then the norm of {displaystyle f} is preserved by this extension. In category-theoretic terms, the field {displaystyle mathbf {K} } is an injective object in the category of locally convex vector spaces. Geometric Hahn–Banach (the Hahn–Banach separation theorems) See also: Hyperplane separation theorem The key element of the Hahn–Banach theorem is fundamentally a result about the separation of two convex sets: {displaystyle {-p(-x-n)-f(n):nin M},} and {displaystyle {p(m+x)-f(m):min 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] Theorem[15] — Let {displaystyle A} and {displaystyle B} be non-empty convex subsets of a real locally convex topological vector space {displaystyle X.} If {displaystyle operatorname {Int} Aneq varnothing } and {displaystyle Bcap operatorname {Int} A=varnothing } then there exists a continuous linear functional {displaystyle f} on {displaystyle X} such that {displaystyle sup f(A)leq inf f(B)} and {displaystyle f(a)0} on {displaystyle U.} Supporting hyperplanes Since points are trivially convex, geometric Hahn-Banach implies that functionals can detect the boundary of a set. In particular, let {displaystyle X} be a real topological vector space and {displaystyle Asubseteq X} be convex with {displaystyle operatorname {Int} Aneq varnothing .} If {displaystyle a_{0}in Asetminus operatorname {Int} A} then there is a functional that is vanishing at {displaystyle a_{0},} but supported on the interior of {displaystyle A.} [15] 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 is Gateaux differentiable at that point.[3] Balanced or disked neighborhoods Let {displaystyle U} be a convex balanced neighborhood of the origin in a locally convex topological vector space {displaystyle X} and suppose {displaystyle xin 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)|leq |f(x)|.} Applications The Hahn–Banach theorem is the first sign of an important philosophy in functional analysis: to understand a space, one should understand its continuous 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: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.

Mazur–Orlicz theorem[3] — Let {displaystyle p:Xto mathbb {R} } be a sublinear function on a real or complex vector space {displaystyle X,} let {displaystyle T} be any set, and let {displaystyle R:Tto mathbb {R} } and {displaystyle v:Tto X} be any maps. The following statements are equivalent: there exists a real-valued linear functional {displaystyle F} on {displaystyle X} such that {displaystyle Fleq p} on {displaystyle X} and {displaystyle Rleq Fcirc v} on {displaystyle T} ; for any finite sequence {displaystyle s_{1},ldots ,s_{n}} of {displaystyle n>0} non-negative real numbers, and any sequence {displaystyle t_{1},ldots ,t_{n}in T} of elements of {displaystyle T,} {displaystyle sum _{i=1}^{n}s_{i}Rleft(t_{i}right)leq pleft(sum _{i=1}^{n}s_{i}vleft(t_{i}right)right).} The following theorem characterizes when any scalar function on {displaystyle X} (not necessarily linear) has a continuous linear extension to all of {displaystyle X.} Theorem (The extension principle[24]) — Let {displaystyle f} a scalar-valued function on a subset {displaystyle S} of a topological vector space {displaystyle X.} Then there exists a continuous linear functional {displaystyle F} on {displaystyle X} extending {displaystyle f} if and only if there exists a continuous seminorm {displaystyle p} on {displaystyle X} such that {displaystyle left|sum _{i=1}^{n}a_{i}f(s_{i})right|leq pleft(sum _{i=1}^{n}a_{i}s_{i}right)} for all positive integers {displaystyle n} and all finite sequences {displaystyle a_{1},ldots ,a_{n}} of scalars and elements {displaystyle s_{1},ldots ,s_{n}} of {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] On the other hand, 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 {displaystyle f} on X such that {displaystyle f(x)neq 0} and {displaystyle f(m)=0} for all {displaystyle min M.} Clearly, the continuous dual space of a TVS X separates points on X if and only if {displaystyle {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. However, 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 (AC). 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). However, 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", that is: a non-constant finitely additive map from {displaystyle B} into {displaystyle [0,1].} (BPI is equivalent to the statement that there are always non-constant probability charges which take only the values 0 and 1.) In ZF, the Hahn–Banach theorem suffices to derive the existence of a non-Lebesgue measurable set.[31] Moreover, 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. In fact, 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, for instance, that {displaystyle F_{b}(x)=F_{b}(0+1x)=f(0)+1b=b} and if {displaystyle min M} then {displaystyle F_{b}(m)=F_{b}(m+0x)=f(m)+0b=f(m).} In fact, if {displaystyle G:Moplus mathbb {R} xto mathbb {R} } is any linear extension of {displaystyle f} to {displaystyle Moplus mathbb {R} x} then {displaystyle G=F_{b}} for {displaystyle b:=G(x).} In other words, every linear extension of {displaystyle f} to {displaystyle Moplus mathbb {R} x} is of the form {displaystyle F_{b}} for some (unique) {displaystyle b.} ^ Explicitly, for any real number {displaystyle bin mathbb {R} ,} {displaystyle F_{b}leq p} on {displaystyle Moplus mathbb {R} x} if and only if {displaystyle aleq bleq c.} Combined with the fact that {displaystyle F_{b}(x)=b,} it follows that the dominated linear extension of {displaystyle f} to {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 {displaystyle f} to {displaystyle Moplus mathbb {R} x} is of the form {displaystyle F_{b}} for some {displaystyle b,} the bounds {displaystyle aleq b=F_{b}(x)leq c} thus also limit the range of possible values (at {displaystyle x} ) that can be taken by any of {displaystyle f} 's dominated linear extensions. Specifically, if {displaystyle F:Xto mathbb {R} } is any linear extension of {displaystyle f} satisfying {displaystyle Fleq p} then for every {displaystyle xin Xsetminus M,} {displaystyle 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=mathbb {R} ^{2},M={(x,0):xin mathbb {R} }.} First, define the simple-minded extension {displaystyle f_{0}(x,y)=f(x),} It doesn't work, since maybe {displaystyle f_{0}leq p} . But it is a step in the right direction. {displaystyle p-f_{0}} is still convex, and {displaystyle p-f_{0}geq f-f_{0}.} Further, {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. If {displaystyle pgeq 0} on {displaystyle mathbb {R} ^{2},} then we are done. Otherwise, pick some {displaystyle vin mathbb {R} ^{2},} such that {displaystyle 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). 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"Hahn–Banach theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] show vte Functional analysis (topics – glossary) show vte Topological vector spaces (TVSs) show vte Banach space topics Categories: Linear algebraTheorems in functional analysisTopological vector spaces

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