Krein–Milman and Banach–Stone
DOI link for Krein–Milman and Banach–Stone
Krein–Milman and Banach–Stone book
If the compact Hausdorff spaces S and T are homeomorphic, the Banach spaces C (T,F, ‖·‖∞) and C (S,F, ‖·‖∞) are linearly isometric. This is easy. The deeper result is the converse, the Banach-Stone theorem, that linear
isometry implies homeomorphism [Theorem 9.6.2]. We prove it using a technique invented by Arens and Kelley. To do that we need a generalization of the notion of vertex of a polygon, something called an extreme point. After developing the elementary properties of extreme points, we prove the KreinMilman theorem which generalizes the notion that if you connect the vertices of a square and fill in the resulting figure, you recover the square. We discuss several variants of the Banach-Stone theorem in Secs. 9.7-9.9.