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 $f:K\to K$ 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
- 1 Rudin, Functional Analysis^{}, Chapter 5.
Title | Schauder fixed point theorem |
---|---|
Canonical name | SchauderFixedPointTheorem |
Date of creation | 2013-03-22 13:45:17 |
Last modified on | 2013-03-22 13:45:17 |
Owner | paolini (1187) |
Last modified by | paolini (1187) |
Numerical id | 12 |
Author | paolini (1187) |
Entry type | Theorem |
Classification | msc 54H25 |
Classification | msc 47H10 |
Classification | msc 46B50 |
Related topic | BrouwerFixedPointTheorem |
Related topic | FixedPoint |
Related topic | TychonoffFixedPointTheorem |