Schauder fixed point theorem

Let X be a normed vector spacePlanetmathPlanetmath, and let KX be a non-empty, compactPlanetmathPlanetmath, and convex set. Then given any continuous mapping f:KK there exists xK 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 TheoremMathworldPlanetmath.

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


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