corollary of Banach-Alaoglu theorem


A Banach spaceMathworldPlanetmath H is isometrically isomorphic to a closed subspace of C(X) for a compactPlanetmathPlanetmath Hausdorff space X.


Let X be the unit ball (*) of *. By the Banach-Alaoglu theorem it is compact in the weak-* topologyMathworldPlanetmath. Define the map Φ:C(X) by (Φf)(φ)=φ(f). This is linear and we have for f:

Φ(f) =supφ(*)|Φ(f)(φ)|=supφ(*)|φ(f)|supφ(*)φff

With the Hahn-Banach theoremMathworldPlanetmath it follows that there is a φ(*) such that φ(f)=f. Thus Φ(f)=f and Φ is an isometric isomorphism, as required. ∎

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