If a normed space $X$ is reflexive, show that $X'$ is reflexive.
Suppose $X$ is reflexive. Then by definition the Canonical mapping $J : X \to X''$ defined by $x \mapsto g_x$ where $g_x(f) = f(x)$ is an isomorphism. We want to show that the mapping $J' : X' \to X'''$ defined by $f \mapsto h_f$ where $h_f(g_x) = g_x(f)$ is an isomorphism. It will suffice to show that $J'$ is onto.
I am unsure about what to do after this. Any help would be greatly appreciated.
- Choose $h \in X'''$, then by definition $h : X'' \to \mathbb R$ is a linear bounded functional.
- Try to find $f \in X'$, that is $f$ such that $f : X \to \mathbb R$ (a linear bounded functional) such that the cannonical mapping maps $f \mapsto h$, i.e., $J'(f) = h(f)$.