corollary of Banach-Alaoglu theorem

Corollary.

A Banach space $\mathrm{H}$ is isometrically isomorphic to a closed subspace of $C\mathit{}\mathrm{(}X\mathrm{)}$ for a compact Hausdorff space $X$.

Proof.

Let $X$ be the unit ball $ℬ({\mathscr{H}}^{*})$ of ${\mathscr{H}}^{*}$. By the Banach-Alaoglu theorem it is compact in the weak-$*$ topology. Define the map $\mathrm{\Phi }:\mathscr{H}\to C(X)$ by $(\mathrm{\Phi }f)(\phi )=\phi (f)$. This is linear and we have for $f\in \mathscr{H}$:

${\parallel \mathrm{\Phi }(f)\parallel }_{\mathrm{\infty }}$$=\underset{\phi \in ℬ({\mathscr{H}}^{*})}{sup}|\mathrm{\Phi }(f)(\phi )|=\underset{\phi \in ℬ({\mathscr{H}}^{*})}{sup}|\phi (f)|\le \underset{\phi \in ℬ({\mathscr{H}}^{*})}{sup}\parallel \phi \parallel \parallel f\parallel \le \parallel f\parallel$

With the Hahn-Banach theorem it follows that there is a $\phi \in ℬ({\mathscr{H}}^{*})$ such that $\phi (f)=\parallel f\parallel$. Thus ${\parallel \mathrm{\Phi }(f)\parallel }_{\mathrm{\infty }}=\parallel f\parallel$ and $\mathrm{\Phi }$ is an isometric isomorphism, as required.∎