# 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 $f\colon K\to K$ 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.

## References

Title Tychonoff fixed point theorem TychonoffFixedPointTheorem 2013-03-22 16:04:11 2013-03-22 16:04:11 paolini (1187) paolini (1187) 8 paolini (1187) Theorem msc 54H25 msc 46B50 msc 47H10 SchauderFixedPointTheorem BrouwerFixedPointTheorem