Editing Contraction mapping (section)

==Locally convex spaces==
In a [[locally convex space]] (''E'',&thinsp;''P'') with [[Topological space|topology]] given by a set ''P'' of [[seminorm]]s, one can define for any ''p''&thinsp;&isin;&thinsp;''P'' a ''p''-contraction as a map ''f'' such that there is some ''k''<sub>''p''</sub> < 1 such that {{nowrap|''p''(''f''(''x'') − ''f''(''y''))}} ≤ {{nowrap|''k<sub>p</sub> p''(''x'' − ''y'')}}. If ''f'' is a ''p''-contraction for all ''p''&thinsp;&isin;&thinsp;''P'' and (''E'',&thinsp;''P'') is sequentially complete, then ''f'' has a fixed point, given as limit of any sequence ''x''<sub>''n''+1</sub> = ''f''(''x''<sub>''n''</sub>), and if (''E'',&thinsp;''P'') is [[Hausdorff space|Hausdorff]], then the fixed point is unique.<ref>{{cite journal |first1=G. L. Jr. |last1=Cain |first2=M. Z. |last2=Nashed |author-link2=Zuhair Nashed |title=Fixed Points and Stability for a Sum of Two Operators in Locally Convex Spaces |journal=Pacific Journal of Mathematics |volume=39 |issue=3 |year=1971 |pages=581–592 |doi=10.2140/pjm.1971.39.581 |doi-access=free }}</ref>