Reflexive 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.

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 or complex numbers), with a norm ${\displaystyle \|\cdot \|}$. Consider its dual normed space ${\displaystyle X'}$, that consists of all continuous linear functionals ${\displaystyle f:X\to {\mathbb {F} }}$ and is equipped with the dual norm ${\displaystyle \|\cdot \|'}$ defined by

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