Fixed-point theorems in infinite-dimensional spaces

In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations.

The first result in the field was the Schauder fixed-point theorem, proved in 1930 by Juliusz Schauder (a previous result in a different vein, the Banach fixed-point theorem for contraction mappings in complete metric spaces was proved in 1922). Quite a number of further results followed. One way in which fixed-point theorems of this kind have had a larger influence on mathematics as a whole has been that one approach is to try to carry over methods of algebraic topology, first proved for finite simplicial complexes, to spaces of infinite dimension. For example, the research of Jean Leray who founded sheaf theory came out of efforts to extend Schauder's work.

Schauder fixed-point theorem: Let C be a nonempty closed convex subset of a Banach space V, if f : C → C is continuous with a compact image, then f has a fixed point.

Tikhonov (Tychonoff) fixed point theorem: Let V be a locally convex topological vector space, for any non-empty compact convex set X in V, any continuous function f : X → X has a fixed point.

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