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 <blockquote>'''[[Schauder fixed-point theorem]]:''' states, in one version, that ifLet ''C'' isbe a [[nonempty]] [[Closed set|closed]] [[Convex set|convex]] subset of a [[Banach space]] ''V'', andif ''f'' is a [[continuous function|continuous map]] from: ''C'' to→ ''C'' whoseis image[[continuous isfunction|continuous]] with a [[compact set|compact]] image, then ''f'' has a fixed point.</blockquote>

The <blockquote>'''Tikhonov (Tychonoff) fixed point theorem:''' isLet applied''V'' tobe anya [[locally convex topological vector space]] ''V''. It states that, for any non-empty compact convex set ''X'' in ''V'', any continuous function ''f'' : ''X'' → ''X'' has a fixed point.</blockquote>

:''&fnof;'':''X'' → ''X'',

has a fixed point.

Other results include the [[Markov–Kakutani fixed-point theorem]] (1936-1938) and the [[Ryll-Nardzewski fixed-point theorem]] (1967) for continuous affine self-mappings of compact convex sets, as well as the [[Earle–Hamilton fixed-point theorem]] (1968) for holomorphic self-mappings of open domains.

<blockquote>'''[[Kakutani fixed point theorem|Kakutani's fixed-point theorem]]:''' statesEvery 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.</blockquote>

: ''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.''

==See also==