The Krein-Milman Theorem

Nirakar Neo's Blog

1. The Krein-Milman theorem in Locally Convex Spaces

My project work this semester focuses to understand the paper the Krein-Milman Theorem in Operator Convexity by Corran Webster and Soren Winkler, which appeared in the Transactions of the AMS [Vol 351, #1, Jan 99, 307-322]. But before reading the paper, it is imperative to understand the (usual) Krein-Milman theorem which is proved in the context of locally convex spaces. My understanding of this part follows the book A Course in Functional Analysis by J B Conway. To begin with we shall collect the preliminaries that we shall need to understand the Krein-Milman theorem.

1.1. Convexity

Let $latex {mathbb{K}}&fg=000000$ denote the real($latex {mathbb{R}}&fg=000000$) or the complex($latex {mathbb{C}}&fg=000000$) number fields. Let $latex {X}&fg=000000$ be a vector space over $latex {mathbb{K}}&fg=000000$. A subset of a vector space is called convex if for any two points in the subset, the line segment joining them…

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