Tychonoff fixed point theorem

Let $X$ be a locally convex topological vector space, and let $K\subset X$ be a non-empty, compact, and convex set. Then given any continuous mapping $f:K\to K$ there exists $x\in K$ such that $f(x)=x$.

Notice that a normed vector space is a locally convex topological vector space so this theorem extends the Schauder fixed point theorem.

Title	Tychonoff fixed point theorem
Canonical name	TychonoffFixedPointTheorem
Date of creation	2013-03-22 16:04:11
Last modified on	2013-03-22 16:04:11
Owner	paolini (1187)
Last modified by	paolini (1187)
Numerical id	8
Author	paolini (1187)
Entry type	Theorem
Classification	msc 54H25
Classification	msc 46B50
Classification	msc 47H10
Related topic	SchauderFixedPointTheorem
Related topic	BrouwerFixedPointTheorem