Reflexive space

In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from ${\displaystyle X}$ into its bidual (which is the strong dual of the strong dual of ${\displaystyle X}$) is a homeomorphism (or equivalently, a TVS isomorphism). A normed space is reflexive if and only if this canonical evaluation map is surjective, in which case this (always linear) evaluation map is an isometric isomorphism and the normed space is a Banach space. Those spaces for which the canonical evaluation map is surjective are called semi-reflexive spaces.

In 1951, R. C. James discovered a Banach space, now known as James' space, that is not reflexive (meaning that the canonical evaluation map is not an isomorphism) but is nevertheless isometrically isomorphic to its bidual (any such isometric isomorphism is necessarily not the canonical evaluation map). So importantly, for a Banach space to be reflexive, it is not enough for it to be isometrically isomorphic to its bidual; it is the canonical evaluation map in particular that has to be a homeomorphism.

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.

Suppose that ${\displaystyle X}$ is a topological vector space (TVS) over the field ${\displaystyle \mathbb {F} }$ (which is either the real or complex numbers) whose continuous dual space, ${\displaystyle X^{\prime },}$ separates points on ${\displaystyle X}$ (that is, for any ${\displaystyle x\in X,x\neq 0}$ there exists some ${\displaystyle x^{\prime }\in X^{\prime }}$ such that ${\displaystyle x^{\prime }(x)\neq 0}$). Let ${\displaystyle X_{b}^{\prime }}$ (some texts write ${\displaystyle X_{\beta }^{\prime }}$) denote the strong dual of ${\displaystyle X,}$ which is the vector space ${\displaystyle X^{\prime }}$ of continuous linear functionals on ${\displaystyle X}$ endowed with the topology of uniform convergence on bounded subsets of ${\displaystyle X}$; 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 ${\displaystyle X}$ is a normed space, then the strong dual of ${\displaystyle X}$ is the continuous dual space ${\displaystyle X^{\prime }}$ with its usual norm topology. The bidual of ${\displaystyle X,}$ denoted by ${\displaystyle X^{\prime \prime },}$ is the strong dual of ${\displaystyle X_{b}^{\prime }}$; that is, it is the space ${\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime }.}$ If ${\displaystyle X}$ is a normed space, then ${\displaystyle X^{\prime \prime }}$ is the continuous dual space of the Banach space ${\displaystyle X_{b}^{\prime }}$ with its usual norm topology.

For any ${\displaystyle x\in X,}$ let ${\displaystyle J_{x}:X^{\prime }\to \mathbb {F} }$ be defined by ${\displaystyle J_{x}\left(x^{\prime }\right)=x^{\prime }(x),}$ where ${\displaystyle J_{x}}$ is a linear map called the evaluation map at ${\displaystyle x}$; since ${\displaystyle J_{x}:X_{b}^{\prime }\to \mathbb {F} }$ is necessarily continuous, it follows that ${\displaystyle J_{x}\in \left(X_{b}^{\prime }\right)^{\prime }.}$ Since ${\displaystyle X^{\prime }}$ separates points on ${\displaystyle X,}$ the linear map ${\displaystyle J:X\to \left(X_{b}^{\prime }\right)^{\prime }}$ defined by ${\displaystyle J(x):=J_{x}}$ is injective where this map is called the evaluation map or the canonical map. Call ${\displaystyle X}$ semi-reflexive if ${\displaystyle J:X\to \left(X_{b}^{\prime }\right)^{\prime }}$ is bijective (or equivalently, surjective) and we call ${\displaystyle X}$ reflexive if in addition ${\displaystyle J:X\to X^{\prime \prime }=\left(X_{b}^{\prime }\right)_{b}^{\prime }}$ 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.

Suppose ${\displaystyle X}$ is a normed vector space over the number field ${\displaystyle \mathbb {F} =\mathbb {R} }$ or ${\displaystyle \mathbb {F} =\mathbb {C} }$ (the real numbers or the complex numbers), with a norm ${\displaystyle \|\,\cdot \,\|.}$ Consider its dual normed space ${\displaystyle X^{\prime },}$ that consists of all continuous linear functionals ${\displaystyle f:X\to \mathbb {F} }$ and is equipped with the dual norm ${\displaystyle \|\,\cdot \,\|^{\prime }}$ defined by $${\displaystyle \|f\|^{\prime }=\sup\{|f(x)|\,:\,x\in X,\ \|x\|=1\}.}$$

The dual ${\displaystyle X^{\prime }}$ is a normed space (a Banach space to be precise), and its dual normed space ${\displaystyle X^{\prime \prime }=\left(X^{\prime }\right)^{\prime }}$ is called bidual space for ${\displaystyle X.}$ The bidual consists of all continuous linear functionals ${\displaystyle h:X^{\prime }\to \mathbb {F} }$ and is equipped with the norm ${\displaystyle \|\,\cdot \,\|^{\prime \prime }}$ dual to ${\displaystyle \|\,\cdot \,\|^{\prime }.}$ Each vector ${\displaystyle x\in X}$ generates a scalar function ${\displaystyle J(x):X^{\prime }\to \mathbb {F} }$ by the formula: $${\displaystyle J(x)(f)=f(x)\qquad {\text{ for all }}f\in X^{\prime },}$$ and ${\displaystyle J(x)}$ is a continuous linear functional on ${\displaystyle X^{\prime },}$ that is, ${\displaystyle J(x)\in X^{\prime \prime }.}$ One obtains in this way a map $${\displaystyle J:X\to X^{\prime \prime }}$$ called evaluation map, that is linear. It follows from the Hahn–Banach theorem that ${\displaystyle J}$ is injective and preserves norms: $${\displaystyle {\text{ for all }}x\in X\qquad \|J(x)\|^{\prime \prime }=\|x\|,}$$ that is, ${\displaystyle J}$ maps ${\displaystyle X}$ isometrically onto its image ${\displaystyle J(X)}$ in ${\displaystyle X^{\prime \prime }.}$ Furthermore, the image ${\displaystyle J(X)}$ is closed in ${\displaystyle X^{\prime \prime },}$ but it need not be equal to ${\displaystyle X^{\prime \prime }.}$

A normed space ${\displaystyle X}$ is called reflexive if it satisfies the following equivalent conditions: