View source for Reflexive space

A [[Banach space]] $X$ that coincides under the canonical imbedding with its second dual $X^{{*}{*}}$ (cf. [[Adjoint space]]). More precisely, let $X^{*}$ be the space dual to $X$, i.e. the set of all continuous linear functionals defined on $X$. If $(x,f)$ is the value of the functional $f \in X^{*}$ on an element $x \in X$, then with a fixed $x$ and $f$ running through $X^{*}$, the formula $(x,f) = \mathcal{F}_x(f)$ defines a linear functional on $X^{*}$, that is, an element of the space $X^{{*}{*}}$. Let $\tilde X \subseteq X^{{*}{*}}$ be the set of such functionals. The correspondence $x \mapsto \mathcal{F}_x$ is an isomorphism which does not change the norm: $\Vert x \Vert = \Vert \mathcal{F}_x \Vert$. If $\tilde X = X^{{*}{*}}$, then the space $X$ is called reflexive. The spaces $\ell_p$ and $L_p(a,b)$, $p>1$, are reflexive, and the space $C[a,b]$ is not reflexive.

A space $X$ is reflexive if and only if the space $X^{*}$ is reflexive. Another criteria of reflexivity of a Banach space $X$ is weak compactness (cf. [[Weak topology]]) of the unit ball of this space.

A reflexive space is weakly complete and a closed subspace of a reflexive space is reflexive.

The concept of reflexivity naturally extends to [[locally convex space]]s.
====References====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  K. Yosida,   "Functional analysis" , Springer  (1980)  pp. Chapt. 8, §1</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Kantorovich,   G.P. Akilov,   "Functional analysis" , Pergamon  (1982)  (Translated from Russian)</TD></TR>
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====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Beauzamy,   "Introduction to Banach spaces and their geometry" , North-Holland  (1982)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  M.M. Day,   "Normed linear spaces" , Springer  (1973)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  D. van Dulst,   "Reflexive and superreflexive Banach spaces" , ''MC Tracts'' , '''102''' , Math. Centre  (1978)</TD></TR>
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