Reflexive Spaces

Show that $(\mathcal{X}^*)^{**}$ and $(\mathcal{X}^{**})^*$ are equal.

Show that for a locally compact space $X$, $C_b(X)$ is reflexive if and only if $X$ is finite.

Let $\mathcal{M} \leq \mathcal{X}$ and let $\rho_{\mathcal{X}} : \mathcal{X} \to \mathcal{X}^{**}$ and $\rho_{\mathcal{M}} : \mathcal{M} \to \mathcal{M}^{**}$ be the natural maps. If $i : \mathcal{M} \to \mathcal{X}$ is the inclusion map, show that there is an isometry $\phi : \mathcal{M}^{**} \to \mathcal{X}^{**}$ such that the diagram

commutes. Prove that $\phi (\mathcal{M}^{**}) = (\mathcal{M}^\perp )^\perp \equiv \left\{ x^{**} \in \mathcal{X}^{**} : x^{**}(\mathcal{M}^\perp ) = 0\right\}$.

Use Exercise 3 to show that if $\mathcal{X}$ is reflexive, then any closed subspace of $\mathcal{X}$ is also reflexive. See Yang [1967].