A normed spaces $(X, \| \cdot \|)$ is called reflexive if the evaluation map $X \to X^{**}$ is an isomorphism. If $X$ is reflexive, it's not analytically distinguishable from it's bidual space $X^{**}$ because they are isometrically isomorphic.
But why does the term "reflexive" make sense? I know that for a relation to be reflexive, every element has to be related to itself, does this notion cary over to normed spaces?
Any help is greatly appreciated.
In general, for a binary relation $R$ on a Cartesian product $X \times X$ of a set with itself to be reflexive means that, for each $x \in X$, the relation $xRx$ holds: $x$ is $R$-related to itself.
So, read "reflexive" as, kind of, "self".
For a normed space $X$, the relation is not on $X \times X$, but on $X \times X^{*}$. A similar relation is on $X^* \times X^{**}$. A composition of these two relations is a relation on $X \times X^{**}$, which is the evaluation map. For this relation to be reflexive, we must have $X$ and $X^{**}$ be algebraically and topologically indistinguishable.