Schauder fixed point theorem

Let $X$ be a normed 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 the unit disc of a finite dimensional vector space is always convex and compact hence this theorem extends Brouwer Fixed Point Theorem.

Notice that the space $X$ is not required to be complete, however the subset $K$ being compact, is complete with respect to the metric induced by $X$.

References

Title Schauder fixed point theorem SchauderFixedPointTheorem 2013-03-22 13:45:17 2013-03-22 13:45:17 paolini (1187) paolini (1187) 12 paolini (1187) Theorem msc 54H25 msc 47H10 msc 46B50 BrouwerFixedPointTheorem FixedPoint TychonoffFixedPointTheorem