In functional analysis, a reflexive space is a Banach space (or more generally a locally convex topological vector space) that coincides with the continuous dual of its continuous dual space, both as linear space and as topological space.
In the area of mathematics known as functional analysis, a reflexive space is a Banach space (or more generally a locally convex topological vector space) that coincides with the continuous dual of its continuous dual space, both as linear space and as topological space. Reflexive Banach spaces are often characterized by their geometric properties.
Let $X$ be a normed space and $X^{\ast \ast}$ denote the second dual vector space of $X$. The canonical map $x \mapsto \hat{x}$ defined by $\hat{x}(f) = f(x), f\in X^{*}$ gives an isometric linear isomorphism (embedding) from $X$ into $X^{**}$. The space $X$ is called reflexive if this map is surjective.
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