In [[mathematics]], a number of '''[[fixed point (mathematics)|fixed-point]] theorems in infinite-dimensional spaces''' generalise the [[Brouwer fixed-point theorem]]. They have applications, for example, to the proof of [[existence theorem]]s for [[partial differential equation]]s.
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]] [[Closed 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|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'', any continuous function
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Other results are the [[Shizuo Kakutani|Kakutani]] and Markov fixed point theorems, as well as the [[Ryll-Nardzewski fixed point theorem]] (1967).
[[Kakutani fixed point theorem|Kakutani's fixed-point theorem]] states that:
: ''Every correspondence that maps a compact convex subset of a locally convex space into itself with a closed graph and convex nonempty images has a fixed point.''
