Schauder fixed point theorem
Let be a normed vector space, and let be a non-empty, compact
, and convex set.Then given any continuous mapping there exists such that .
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 is not required to be complete, however the subset being compact,is complete with respect to the metric induced by .
References
- 1 Rudin, Functional Analysis
, Chapter 5.