corollary of Banach-Alaoglu theorem

Corollary.

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

Proof.

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

 $\displaystyle\|\Phi(f)\|_{\infty}$ $\displaystyle=\sup_{\varphi\in\mathcal{B}(\mathscr{H}^{*})}|\Phi(f)(\varphi)|=% \sup_{\varphi\in\mathcal{B}(\mathscr{H}^{*})}|\varphi(f)|\leq\sup_{\varphi\in% \mathcal{B}(\mathscr{H}^{*})}\|\varphi\|\|f\|\leq\|f\|$

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

Title corollary of Banach-Alaoglu theorem CorollaryOfBanachAlaogluTheorem 2013-03-22 18:34:45 2013-03-22 18:34:45 karstenb (16623) karstenb (16623) 4 karstenb (16623) Corollary msc 46B10