On Choquet Theorem And Application

This thesis is about the proofs of Krein-Milman theorem,Choquet theorem and their applications. By studying some properties about the compact convex set and endpoint in the locally convex space, we comb the proofs [6] about Choquet theorem and Krein-Milman theorem. The proven process is better and more concise. Lastly, we give an integral representation theorem of a completely bounded function by Choquet theorem. The structure of this thesis goes as follow:In the introduction, we recall some background knowledge. We also describe the problem we will study.In chapter one, We recall some necessary basic knowledge and important basic lemma for solving the problem.In chapter two, we reprove some lemmas on the locally convex space by the methods of contradiction, structuring the continuous map. Then using those lem-mas, we reprove Krein-Milman theorem and Choquet theorem in more details.In chapter three, we structure some examples to show the applications of Minkowski theorem and Choquet theorem.