Every nonempty, compact convex set $M$ in a locally convex space has fixed point property

Question

I need to prove that "Every nonempty, compact convex set $M$ in a locally convex space has fixed point property".

In the book the reference has been given to "Eisenack & Frenske, 1944, page 44". I am unable to find the book. Also how do I proceed? I can try proving if the problem can be divided into smaller parts as well.

Thanks for the help!

Answer 1

The result you are looking for is called Tychonoff's fixed point theorem, a generalization of Schauder's theorem for locally convex spaces.

For a proof of this result you can look at Eberhard Zeidler "Nonlinear functional analysis and its applications, Vol. 1: Fixed-point theorems", Springer, New York, 1986. But knowing the name you might just be able to google and find other references online.