[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki12\/reflexive-space-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki12\/reflexive-space-wikipedia\/","headline":"Reflexive space – Wikipedia","name":"Reflexive space – Wikipedia","description":"Locally convex topological vector space In the area of mathematics known as functional analysis, a reflexive space is a locally","datePublished":"2020-03-28","dateModified":"2020-03-28","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki12\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki12\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/68baa052181f707c662844a465bfeeb135e82bab","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/68baa052181f707c662844a465bfeeb135e82bab","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki12\/reflexive-space-wikipedia\/","wordCount":44528,"articleBody":"Locally convex topological vector spaceIn the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X{displaystyle X} into its bidual (which is the strong dual of the strong dual of X{displaystyle X}) is an isomorphism of TVSs.Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) X{displaystyle X} is reflexive if and only if the canonical evaluation map from X{displaystyle X} into its bidual is surjective;in this case the normed space is necessarily also a Banach space.In 1951, R. C. James discovered a Banach space, now known as James’ space, that is not reflexive but is nevertheless isometrically isomorphic to its bidual (any such isomorphism is thus necessarily not the canonical evaluation map).Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular.  Hilbert spaces are prominent examples of reflexive Banach spaces.  Reflexive Banach spaces are often characterized by their geometric properties.Definition[edit]Definition of the bidualSuppose that X{displaystyle X} is a topological vector space (TVS) over the field F{displaystyle mathbb {F} } (which is either the real or complex numbers) whose continuous dual space, X\u2032,{displaystyle X^{prime },} separates points on X{displaystyle X} (that is, for any x\u2208X,x\u22600{displaystyle xin X,xneq 0} there exists some x\u2032\u2208X\u2032{displaystyle x^{prime }in X^{prime }} such that x\u2032(x)\u22600{displaystyle x^{prime }(x)neq 0}).Let Xb\u2032{displaystyle X_{b}^{prime }} and Xb\u2032{displaystyle X_{b}^{prime }} both denote the strong dual of X,{displaystyle X,} which is the vector space X\u2032{displaystyle X^{prime }} of continuous linear functionals on X{displaystyle X} endowed with the topology of uniform convergence on bounded subsets of X{displaystyle X};this topology is also called the strong dual topology and it is the “default” topology placed on a continuous dual space (unless another topology is specified).If X{displaystyle X} is a normed space, then the strong dual of X{displaystyle X} is the continuous dual space X\u2032{displaystyle X^{prime }} with its usual norm topology.The bidual of X,{displaystyle X,} denoted by X\u2032\u2032,{displaystyle X^{prime prime },} is the strong dual of Xb\u2032{displaystyle X_{b}^{prime }}; that is, it is the space (Xb\u2032)b\u2032.{displaystyle left(X_{b}^{prime }right)_{b}^{prime }.}If X{displaystyle X} is a normed space, then X\u2032\u2032{displaystyle X^{prime prime }} is the continuous dual space of the Banach space Xb\u2032{displaystyle X_{b}^{prime }} with its usual norm topology.Definitions of the evaluation map and reflexive spacesFor any x\u2208X,{displaystyle xin X,} let Jx:X\u2032\u2192F{displaystyle J_{x}:X^{prime }to mathbb {F} } be defined by Jx(x\u2032)=x\u2032(x),{displaystyle J_{x}left(x^{prime }right)=x^{prime }(x),} where Jx{displaystyle J_{x}} is a linear map called the evaluation map at x{displaystyle x};since Jx:Xb\u2032\u2192F{displaystyle J_{x}:X_{b}^{prime }to mathbb {F} } is necessarily continuous, it follows that Jx\u2208(Xb\u2032)\u2032.{displaystyle J_{x}in left(X_{b}^{prime }right)^{prime }.}Since X\u2032{displaystyle X^{prime }} separates points on X,{displaystyle X,} the linear map J:X\u2192(Xb\u2032)\u2032{displaystyle J:Xto left(X_{b}^{prime }right)^{prime }} defined by J(x):=Jx{displaystyle J(x):=J_{x}} is injective where this map is called the evaluation map or the canonical map.Call X{displaystyle X} semi-reflexive if J:X\u2192(Xb\u2032)\u2032{displaystyle J:Xto left(X_{b}^{prime }right)^{prime }} is bijective (or equivalently, surjective) and we call X{displaystyle X} reflexive if in addition J:X\u2192X\u2032\u2032=(Xb\u2032)b\u2032{displaystyle J:Xto X^{prime prime }=left(X_{b}^{prime }right)_{b}^{prime }} is an isomorphism of TVSs.A normable space is reflexive if and only if it is semi-reflexive or equivalently, if and only if the evaluation map is surjective.Reflexive Banach spaces[edit]Suppose X{displaystyle X} is a normed vector space over the number field F=R{displaystyle mathbb {F} =mathbb {R} } or F=C{displaystyle mathbb {F} =mathbb {C} } (the real numbers or the complex numbers), with a norm \u2016\u22c5\u2016.{displaystyle |,cdot ,|.} Consider its dual normed space X\u2032,{displaystyle X^{prime },} that consists of all continuous linear functionals f:X\u2192F{displaystyle f:Xto mathbb {F} } and is equipped with the dual norm \u2016\u22c5\u2016\u2032{displaystyle |,cdot ,|^{prime }} defined by\u2016f\u2016\u2032=sup{|f(x)|:x\u2208X,\u00a0\u2016x\u2016=1}.{displaystyle |f|^{prime }=sup{|f(x)|,:,xin X, |x|=1}.}The dual X\u2032{displaystyle X^{prime }} is a normed space (a Banach space to be precise), and its dual normed space X\u2032\u2032=(X\u2032)\u2032{displaystyle X^{prime prime }=left(X^{prime }right)^{prime }} is called bidual space for X.{displaystyle X.} The bidual consists of all continuous linear functionals h:X\u2032\u2192F{displaystyle h:X^{prime }to mathbb {F} } and is equipped with the norm \u2016\u22c5\u2016\u2032\u2032{displaystyle |,cdot ,|^{prime prime }} dual to \u2016\u22c5\u2016\u2032.{displaystyle |,cdot ,|^{prime }.} Each vector x\u2208X{displaystyle xin X} generates a scalar function J(x):X\u2032\u2192F{displaystyle J(x):X^{prime }to mathbb {F} } by the formula:J(x)(f)=f(x)\u00a0for all\u00a0f\u2208X\u2032,{displaystyle J(x)(f)=f(x)qquad {text{ for all }}fin X^{prime },}and J(x){displaystyle J(x)} is a continuous linear functional on X\u2032,{displaystyle X^{prime },} that is, J(x)\u2208X\u2032\u2032.{displaystyle J(x)in X^{prime prime }.}  One obtains in this way a mapJ:X\u2192X\u2032\u2032{displaystyle J:Xto X^{prime prime }}called evaluation map, that is linear. It follows from the Hahn\u2013Banach theorem that J{displaystyle J} is injective and preserves norms:\u00a0for all\u00a0x\u2208X\u2016J(x)\u2016\u2032\u2032=\u2016x\u2016,{displaystyle {text{ for all }}xin Xqquad |J(x)|^{prime prime }=|x|,}that is, J{displaystyle J} maps X{displaystyle X} isometrically onto its image J(X){displaystyle J(X)} in X\u2032\u2032.{displaystyle X^{prime prime }.}  Furthermore, the image J(X){displaystyle J(X)} is closed in X\u2032\u2032,{displaystyle X^{prime prime },} but it need not be equal to X\u2032\u2032.{displaystyle X^{prime prime }.}A normed space X{displaystyle X} is called reflexive if it satisfies the following equivalent conditions: the evaluation map J:X\u2192X\u2032\u2032{displaystyle J:Xto X^{prime prime }} is surjective, the evaluation map J:X\u2192X\u2032\u2032{displaystyle J:Xto X^{prime prime }} is an isometric isomorphism of normed spaces, the evaluation map J:X\u2192X\u2032\u2032{displaystyle J:Xto X^{prime prime }} is an isomorphism of normed spaces.A reflexive space X{displaystyle X} is a Banach space, since X{displaystyle X} is then isometric to the Banach space X\u2032\u2032.{displaystyle X^{prime prime }.}[edit]A Banach space X{displaystyle X} is reflexive if it is linearly isometric to its bidual under this canonical embedding J.{displaystyle J.} James’ space is an example of a non-reflexive space which is linearly isometric to its bidual. Furthermore, the image of James’ space under the canonical embedding J{displaystyle J} has codimension one in its bidual.[2]A Banach space X{displaystyle X} is called quasi-reflexive (of order d{displaystyle d}) if the quotient X\u2032\u2032\/J(X){displaystyle X^{prime prime }\/J(X)} has finite dimension d.{displaystyle d.}Examples[edit]Every finite-dimensional normed space is reflexive, simply because in this case, the space, its dual and bidual all have the same linear dimension, hence the linear injection J{displaystyle J} from the definition is bijective, by the rank\u2013nullity theorem.The Banach space c0{displaystyle c_{0}} of scalar sequences tending to 0 at infinity, equipped with the supremum norm, is not reflexive.  It follows from the general properties below that \u21131{displaystyle ell ^{1}} and \u2113\u221e{displaystyle ell ^{infty }} are not reflexive, because \u21131{displaystyle ell ^{1}} is isomorphic to the dual of c0{displaystyle c_{0}} and \u2113\u221e{displaystyle ell ^{infty }} is isomorphic to the dual of \u21131.{displaystyle ell ^{1}.}All Hilbert spaces are reflexive, as are the Lp spaces Lp{displaystyle L^{p}} for 1){displaystyle L^{1}(mu )} and L\u221e(\u03bc){displaystyle L^{infty }(mu )} spaces are not reflexive (unless they are finite dimensional, which happens for example when \u03bc{displaystyle mu } is a measure on a finite set).  Likewise, the Banach space C([0,1]){displaystyle C([0,1])} of continuous functions on [0,1]{displaystyle [0,1]} is not reflexive.The spaces Sp(H){displaystyle S_{p}(H)} of operators in the Schatten class  on a Hilbert space H{displaystyle H} are uniformly convex, hence reflexive, when 11,{displaystyle ell ^{1},} and S\u221e(H)=L(H){displaystyle S_{infty }(H)=L(H)} (the bounded linear operators on H{displaystyle H}) is not reflexive, because it contains a subspace isomorphic to \u2113\u221e.{displaystyle ell ^{infty }.}  In both cases, the subspace can be chosen to be the operators diagonal with respect to a given orthonormal basis of H.{displaystyle H.}Properties[edit]If a Banach space Y{displaystyle Y} is isomorphic to a reflexive Banach space X{displaystyle X} then Y{displaystyle Y} is reflexive.[3]Every closed linear subspace of a reflexive space is reflexive.  The continuous dual of a reflexive space is reflexive.  Every quotient of a reflexive space by a closed subspace is reflexive.[4]Let X{displaystyle X} be a Banach space. The following are equivalent.The space X{displaystyle X} is reflexive.The continuous dual of X{displaystyle X} is reflexive.[5]The closed unit ball of X{displaystyle X} is compact in the weak topology. (This is known as Kakutani’s Theorem.)Every bounded sequence in X{displaystyle X} has a weakly convergent subsequence.[7]Every continuous linear functional on X{displaystyle X} attains its supremum on the closed unit ball in X.{displaystyle X.}[8] (James’ theorem)Since norm-closed convex subsets in a Banach space are weakly closed,[9]it follows from the third property that closed bounded convex subsets of a reflexive space X{displaystyle X} are weakly compact.  Thus, for every decreasing sequence of non-empty closed bounded convex subsets of X,{displaystyle X,} the intersection is non-empty.  As a consequence, every continuous convex function f{displaystyle f} on a closed convex subset C{displaystyle C} of X,{displaystyle X,} such that the setCt={x\u2208C:f(x)\u2264t}{displaystyle C_{t}={xin C,:,f(x)leq t}}is non-empty and bounded for some real number t,{displaystyle t,} attains its minimum value on C.{displaystyle C.}The promised geometric property of reflexive Banach spaces is the following: if C{displaystyle C} is a closed non-empty convex subset of the reflexive space X,{displaystyle X,} then for every x\u2208X{displaystyle xin X} there exists a c\u2208C{displaystyle cin C} such that \u2016x\u2212c\u2016{displaystyle |x-c|} minimizes the distance between x{displaystyle x} and points of C.{displaystyle C.}  This follows from the preceding result for convex functions, applied tof(y)+\u2016y\u2212x\u2016.{displaystyle f(y)+|y-x|.}  Note that while the minimal distance between x{displaystyle x} and C{displaystyle C} is uniquely defined by x,{displaystyle x,} the point c{displaystyle c} is not.  The closest point c{displaystyle c} is unique when X{displaystyle X} is uniformly convex.A reflexive Banach space is separable if and only if its continuous dual is separable.  This follows from the fact that for every normed space Y,{displaystyle Y,} separability of the continuous dual Y\u2032{displaystyle Y^{prime }} implies separability of Y.{displaystyle Y.}[10]Super-reflexive space[edit]Informally, a super-reflexive Banach space X{displaystyle X} has the following property: given an arbitrary Banach space Y,{displaystyle Y,} if all finite-dimensional subspaces of Y{displaystyle Y} have a very similar copy sitting somewhere in X,{displaystyle X,} then Y{displaystyle Y} must be reflexive.  By this definition, the space X{displaystyle X} itself must be reflexive.  As an elementary example, every Banach space Y{displaystyle Y} whose two dimensional subspaces are isometric to subspaces of X=\u21132{displaystyle X=ell ^{2}} satisfies the parallelogram law, hence[11]Y{displaystyle Y} is a Hilbert space, therefore Y{displaystyle Y} is reflexive.  So \u21132{displaystyle ell ^{2}} is super-reflexive.The formal definition does not use isometries, but almost isometries.  A Banach space Y{displaystyle Y} is finitely representable[12]in a Banach space X{displaystyle X} if for every finite-dimensional subspace Y0{displaystyle Y_{0}} of Y{displaystyle Y} and every "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki12\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki12\/reflexive-space-wikipedia\/#breadcrumbitem","name":"Reflexive space – Wikipedia"}}]}]