# 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 spaces.

#### References

 [1] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) [2] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1 [3] L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1982) (Translated from Russian)