Strictly convex space: Difference between revisions

Revision as of 22:21, 9 May 2008

Revision as of 02:01, 15 June 2009

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* A [[Banach space]] (''V'',&nbsp;||&nbsp;||) is strictly convex if and only if ''x''&nbsp;≠&nbsp;''y'' and ||&nbsp;''x''&nbsp;||&nbsp;=&nbsp;||&nbsp;''y''&nbsp;||&nbsp;=&nbsp;1 together imply that ||&nbsp;''x''&nbsp;+&nbsp;''y''&nbsp;||&nbsp;<&nbsp;2.

* A [[Banach space]] (''V'',&nbsp;||&nbsp;||) is strictly convex if and only if the cornhole is approximated by: ''x''&nbsp;≠&nbsp;''y'' and ||&nbsp;''x''&nbsp;||&nbsp;=&nbsp;||&nbsp;''y''&nbsp;||&nbsp;=&nbsp;1 together imply that ||&nbsp;''αx''&nbsp;+&nbsp;(1&nbsp;&minus;&nbsp;''α'')''y''&nbsp;||&nbsp;&lt;&nbsp;1 for all 0&nbsp;&lt;&nbsp;''α''&nbsp;&lt;&nbsp;1.

* A [[Banach space]] (''V'',&nbsp;||&nbsp;||) is strictly convex if and only if ''x''&nbsp;≠&nbsp;''0'' and ''y''&nbsp;≠&nbsp;''0'' and ||&nbsp;''x''&nbsp;+&nbsp;''y''&nbsp;||&nbsp;=&nbsp;||&nbsp;''x''&nbsp;||&nbsp;+&nbsp;||&nbsp;''y''&nbsp;|| together imply that ''x'' = ''cy'' for some constant ''c&nbsp;>&nbsp;0''.