# 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 $\mathcal{B}({\mathscr{H}}^{*})$ 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 \mathcal{B}({\mathscr{H}}^{*})}{sup}|\mathrm{\Phi}(f)(\phi )|=\underset{\phi \in \mathcal{B}({\mathscr{H}}^{*})}{sup}|\phi (f)|\le \underset{\phi \in \mathcal{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 $\phi \in \mathcal{B}({\mathscr{H}}^{*})$ 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.
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Title | corollary of Banach-Alaoglu theorem |
---|---|

Canonical name | CorollaryOfBanachAlaogluTheorem |

Date of creation | 2013-03-22 18:34:45 |

Last modified on | 2013-03-22 18:34:45 |

Owner | karstenb (16623) |

Last modified by | karstenb (16623) |

Numerical id | 4 |

Author | karstenb (16623) |

Entry type | Corollary |

Classification | msc 46B10 |