Banach–Stone theorem

The Banach -Stone is a classic mathematical theorem, which is located in the transition area between topology and functional analysis. He goes back to the two mathematicians Stefan Banach and Marshall Stone. The statement of the theorem can be summarized so that the structure of a compact Hausdorff space and the structure of the corresponding Banach space of continuous real-valued functions given to him are directly linked to each other and define mutually up to isomorphism.

The Banach - Stone is the subject matter and the starting point of a series of further investigations.

Wording of the sentence

Given two compact Hausdorff spaces and plus the associated - Banach spaces of continuous real functions and, in each case provided with the supremum norm.

Then: and are homeomorphic if and only if and as Banach spaces are isometrically isomorphic.

Note

The statement of the theorem is true in the same way, when one considers the corresponding Funktionenbanachräume of continuous complex-valued functions instead of continuous real-valued functions of Funktionenbanachräume.

For the proof

The main part of the proof is the demonstration that the isometric isomorphism of the two Funktionenbanachräume pulls the homeomorphism of the underlying compact Hausdorff spaces by themselves, because the proof of the reverse implication is easy. However, low-lying aids of topology and functional analysis are required for this part of evidence, in particular, the following theorems:

• Krein - Milman
• Banach - Alaoglu
• Rieszscher representation theorem
• Lemma of Urysohn

It should be noted that the proof of the theorem of Krein - Milman and is thus also the proof of the theorem of Banach -Stone lemma anger (or an equivalent maximum principle of set theory ) is used. A detailed presentation of proof can be found in Ronald Larsen's book Functional Analysis.