A surjective linear isometry between compact Hausdorff function spaces has the form , where is a homeomorphism and is continuous with modulus one. Thus the Banach space structure of determines up to homeomorphism.
For complex on a compact Hausdorff space, the extreme points of its dual unit ball are precisely unimodular multiples of Dirac measures. A measure whose variation measure has mass on two disjoint sets splits as a nontrivial convex combination of normalized restrictions and is not extreme. This description is the key to the Banach–Stone theorem.
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The Banach–Stone theorem is a fundamental result in functional analysis that provides a characterization of certain types of continuous linear operators between spaces of continuous functions. Specifically, it deals with the relationship between spaces of continuous functions on compact Hausdorff spaces.