Content deleted Content added
No edit summary |
put category at the bottom, thus removing an ugly space from the top. |
||
Line 1:
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 research of [[Jean Leray]] that proved influential for [[algebraic topology]] and [[sheaf theory]] was motivated by the need to go beyond the
[[Category:Functional analysis]]▼
[[Category:Theorems]]▼
▲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 research of [[Jean Leray]] that proved influential for [[algebraic topology]] and [[sheaf theory]] was motivated by the need to go beyond the '''Schauder fixed point theorem''', proved in 1930 by [[Juliusz Schauder]].
The '''Schauder fixed point theorem''' states, in one version, that if ''C'' is a [[nonempty]] [[closed set|closed]] [[convex]] subset of a [[Banach space]] ''V'' and ''f'' is a continuous map from ''C'' to ''C'' whose image is [[compact|countably compact]], then ''f'' has a fixed point.
Line 13 ⟶ 10:
Other results are the Kakutani and Markov fixed point theorems, now subsumed in the Ryll-Nardzewski fixed point theorem (1967).
▲[[Category:Functional analysis]]
▲[[Category:Theorems]]
| |||