75.85.121.0 at 04:34, 16 December 2014

2014-12-16T04:34:34Z

en>Monkbot: /* Proof of irrationality */Fix CS1 deprecated date parameter errors

2014-01-30T08:04:39Z

Proof of irrationality: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 10:04, 30 January 2014
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	The ~~title~~ of the ~~author~~ is ~~Figures~~. ~~Years~~ in ~~the past we moved~~ to ~~North Dakota~~. ~~What I love performing~~ is ~~doing ceramics but I haven~~'~~t made a dime with it~~. ~~Managing individuals~~ is ~~what I do~~ and ~~the wage has been truly fulfilling~~.<br><br>~~Review my web site: [http:~~//~~www~~.~~sex~~-~~porn~~-~~tube~~.ch/~~blog~~/~~36870 http:~~/~~/www~~.~~sex~~-~~porn~~-~~tube~~.ch]		In [[mathematics]], '''uniformly convex spaces''' (or '''uniformly rotund spaces''') are common examples of [[reflexive space\|reflexive]] [[Banach space]]s. The concept of uniform convexity was first introduced by [[James A. Clarkson]] in 1936.

			== Definition ==
			A '''uniformly convex space''' is a [[normed vector space]] so that, for every <math>\epsilon>0</math> there is some <math>\delta>0</math> so that for any two vectors with <math>\\|x\\| = 1</math> and <math>\\|y\\| = 1,</math> the condition

			:<math>\\|x-y\\|\geq \epsilon</math>

			implies that:

			:<math>\left\\|\frac{x+y}{2}\right\\| \leq 1-\delta.</math>

			Intuitively, the center of a line segment inside the [[unit ball]] must lie deep inside the unit ball unless the segment is short.

			== Properties ==
			* The [[Milman–Pettis theorem]] states that every uniformly convex [[Banach space]] is [[reflexive space\|reflexive]], while the converse is not true.
			* If <math> \{f_n\}_{n=1}^{\infty} </math> is a sequence in a uniformly convex Banach space which converges weakly to <math> f </math> and satisfies <math> \\|f_n\\| \to \\|f\\|,</math> then <math> f_n </math> converges strongly to <math> f </math>, that is, <math> \\|f_n - f\\| \to 0 </math>.
			* A Banach space <math> X </math> is uniformly convex if and only if its dual <math> X^* </math> is [[uniformly smooth space\|uniformly smooth]].
			* Every uniformly convex space is [[strictly convex space\|strictly convex]].

			== Examples ==

			* Every Hilbert space is uniformly convex.
			* Every closed subspace of a uniformly convex Banach space is uniformly convex.
			* [[Hanner's inequalities]] imply that [[lp space\|L<sup>''p''</sup> spaces]] <math>(1<p<\infty)</math> are uniformly convex.
			* Conversely, <math>L^\infty</math> is not uniformly convex. For example, in <math>\mathbb{R}^2</math> consider <math>x=(1,1)</math> and <math>y=(0,1)</math>. Then <math>\\|x\\|_{\infty}=\\|y\\|_{\infty}=1</math> and <math>\\|x+y\\|_{\infty}=\\|(1,2)\\|_{\infty}=2</math>, but <math>\\|x-y\\|_{\infty}=\\|(1,0)\\|_{\infty}=1</math>.

			==See also==

			* [[Modulus and characteristic of convexity]]
			* [[Convex_function#Uniformly_convex_functions\|Uniformly convex function]]
			* [[Uniformly smooth space]]

			== References ==
			* {{Cite journal\|first=J. A.\|last=Clarkson\|title=Uniformly convex spaces\|journal=Trans. Amer. Math. Soc.\|volume=40\|year=1936\|pages=396–414\|doi=10.2307/1989630\|jstor=1989630\|issue=3\|publisher=American Mathematical Society\|postscript=<!--None-->}}.
			* {{Cite journal\|first=O.\|last=Hanner\|authorlink=Olof Hanner\|title=On the uniform convexity of <math>L^p</math> and <math>l^p</math>\|journal=Ark. Mat.\|volume=3\|year=1956\|pages=239–244\|postscript=<!--None-->\|doi=10.1007/BF02589410}}.

			*{{cite book
			\|author=Beauzamy, Bernard
			\|title=Introduction to Banach Spaces and their Geometry
			\|year=1985 [1982]
			\|edition=Second revised
			\|publisher=North-Holland
			\|isbn=0-444-86416-4
			}}
			* {{cite journal
			\|doi=10.1007/BF02762802
			\|author=[[Per Enflo]]
			\|title=Banach spaces which can be given an equivalent uniformly convex norm
			\|journal=Israel Journal of Mathematics
			\|volume=13
			\|issue=3–4
			\|year=1972
			\|pages=281–288}}
			* [[Joram Lindenstrauss\|Lindenstrauss, Joram]] and Benyamini, Yoav. ''Geometric nonlinear functional analysis'' Colloquium publications, 48. American Mathematical Society.

			[[Category:Convex analysis]]
			[[Category:Banach spaces]]

78.223.12.118 at 17:03, 15 August 2012

2012-08-15T17:03:09Z

New page

The title of the author is Figures. Years in the past we moved to North Dakota. What I love performing is doing ceramics but I haven't made a dime with it. Managing individuals is what I do and the wage has been truly fulfilling.<br><br>Review my web site: [http://www.sex-porn-tube.ch/blog/36870 http://www.sex-porn-tube.ch]

Square root of 3 - Revision history

75.85.121.0 at 04:34, 16 December 2014

en>Monkbot: /* Proof of irrationality */Fix CS1 deprecated date parameter errors

78.223.12.118 at 17:03, 15 August 2012