Fixed-point theorems in infinite-dimensional spaces: Difference between revisions

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The first result in the field was the '''Schauder fixed point theorem''', proved in 1930 by [[Juliusz Schauder]]. 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 complex]]es, 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.

The '''Schauder fixed point theorem''' states, in one version, that if ''C'' is a [[nonempty]] [[closedClosed set|closed]] [[Convex set|convex]] subset of a [[Banach space]] ''V'' and ''f'' is a [[continuous function|continuous map]] from ''C'' to ''C'' whose image is [[compact set|countably compact]], then ''f'' has a fixed point.

The '''Tikhonov (Tychonoff) fixed point theorem''' is applied to any [[locally convex topological vector space]] ''V''. It states that for any non-empty compact convex set ''X'' in ''V'', and continuous function