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^{}.
References
- 1 Rudin, Functional Analysis^{}, Chapter 5.
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 |