Uniformly convex space

Uniformly convex spaces are a special class of normalized spaces considered in mathematics . These spaces were introduced in 1936 by James A. Clarkson by means of a geometric property of the unit sphere. The uniformly convex Banach spaces are reflexive and have an important property for approximation theory .

Motivation and Definition

The midpoint between and lies in the case of the Euclidean norm, not in the case of the sum norm. ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$

Since the unit sphere of a normalized space is convex , the midpoint between two vectors and the unit sphere is again in the unit sphere. We investigate the distance of such a center from the edge of the unit sphere. ${\displaystyle \{x\in E;\|x\|\leq 1\}}$ ${\displaystyle E}$ ${\displaystyle {\tfrac {1}{2}}(x+y)}$ ${\displaystyle x}$ ${\displaystyle y}$

Considering on the Euclidean norm , then the unit sphere of the unit circle in the plane. If the center of two edge points is formed, then this center point lies further inside the circle, the further the two edge points are from each other. ${\displaystyle {\mathbb {R} }^{2}}$

If, on the other hand, one looks at the sum norm defined by , then the 'unit sphere' is a square. It applies to apparently , , and . Although the two edge points and are very far apart, their center point is still on the edge of the unit sphere. ${\displaystyle {\mathbb {R} }^{2}}$ ${\displaystyle \|(x,y)\|_{1}:=|x|+|y|}$ ${\displaystyle e_{1}:=(1,0),e_{2}:=(0,1)}$ ${\displaystyle \|e_{1}\|_{1}=1}$ ${\displaystyle \|e_{2}\|_{1}=1}$ ${\displaystyle \|{\tfrac {1}{2}}(e_{1}+e_{2})\|_{1}=1}$ ${\displaystyle \|e_{1}-e_{2}\|=2}$ ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$

It is therefore a special geometrical property that two vectors of the unit sphere must be close to each other if their center is close to the edge. Therefore one defines:

A normed space is uniformly convex , if for every one there, so the following applies: If using , and so follows . ${\displaystyle E}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle \delta >0}$ ${\displaystyle x,y\in E}$ ${\displaystyle \|x\|\leq 1}$ ${\displaystyle \|y\|\leq 1}$ ${\displaystyle \|{\tfrac {1}{2}}(x+y)\|>1-\delta }$ ${\displaystyle \|x-y\|<\varepsilon }$

This is a property of the norm . If one goes over to an equivalent norm , this property can be lost, as the two examples considered at the beginning show.

Examples

• It is easy to show by means of the parallelogram equation that interior product spaces are uniformly convex.

• YES Clarkson has this property for the Banach L ^p [0,1] , detected ( set of Clarkson ). A simpler proof emerged as a consequence of the Hanner inequalities proved by Olof Hanner in 1956 . This statement was substantially generalized in 1950 by EJ McShane . If the space is uniformly convex, any positive measure , then uniformly convex is also . It is the Banach space of equivalence classes of measurable functions with values in so . ${\displaystyle 1<p<\infty }$ ${\displaystyle E}$ ${\displaystyle \mu }$ ${\displaystyle 1<p<\infty }$ ${\displaystyle L^{p}(\mu ,E)}$ ${\displaystyle L^{p}(\mu ,E)}$ ${\displaystyle f}$ ${\displaystyle E}$ ${\displaystyle \int \|f(\cdot )\|^{p}d\mu <\infty }$

• 1967 CA McCarthy has the uniform convexity for shadow classes with proven. ${\displaystyle 1<p<\infty }$

Milman's Theorem

David Milman has demonstrated the following important property of uniformly convex spaces:

Milman's theorem : Uniformly convex Banach spaces are reflexive.

This result was also found independently of Milman by Billy James Pettis (1913–1979), which is why one sometimes speaks of the Milman-Pettis theorem . The class of uniformly convex spaces is really smaller than the class of reflexive spaces, because there are reflexive Banach spaces that are not isomorphic to uniformly convex spaces.

One can even show that uniformly convex Banach spaces have the Banach-Saks property (a theorem by S. Kakutani ), and that Banach spaces with the Banach-Saks property are reflexive (a theorem by T. Nishiura and D. Waterman ).

The approximation theorem

The following statements, which are also known as the approximation theorem, show the importance of the uniformly convex spaces for approximation theory. Many approximation problems can be reformulated in such a way that a vector can be found in a convex set (e.g. in a subspace) which has the shortest distance to a given vector. The following statements for a real normed space apply , and a completed and convex subset : ${\displaystyle E}$ ${\displaystyle x\in E}$ ${\displaystyle Y\subset E}$

• Uniqueness: If strictly convex , there is at most one with . ${\displaystyle E}$ ${\displaystyle y_{0}\in Y}$ ${\displaystyle \|x-y_{0}\|=\inf _{y\in Y}\|x-y\|}$
• Existence: If there is a uniformly convex Banach space, then there is a (according to the above uniquely determined) with . (Note that evenly convex spaces are strictly convex.) ${\displaystyle E}$ ${\displaystyle y(x)\in Y}$ ${\displaystyle \|x-y_{0}\|=\inf _{y\in Y}\|x-y\|}$
• Aspect of continuity: If there is a uniformly convex Banach space and a normalized subspace in closed , then the proximum mapping that assigns the (previously described) to each is continuous . ${\displaystyle E}$ ${\displaystyle Y\subset E}$ ${\displaystyle E}$ ${\displaystyle x\mapsto y(x)}$ ${\displaystyle x\in E}$ ${\displaystyle y(x)\in Y}$

Convexity module

You bet for a number ${\displaystyle 0\leq \alpha \leq 2}$

${\displaystyle \delta _{E}(\alpha ):=\inf\{1-{\frac {1}{2}}\|x+y\|;\,x,y\in E,\|x\|\leq 1,\|y\|\leq 1,\|x-y\|=\alpha \}}$

and calls the function defined thereby the convexity modulus of . For uniformly convex spaces, by definition, applies to all , and it can be shown that the convexity module is a monotonic function, even the mapping is monotonic. A theorem by MI Kadec represents a necessary condition for the unconditional convergence of series in uniformly convex spaces: ${\displaystyle \delta _{E}:[0,2]\rightarrow [0,1]}$ ${\displaystyle E}$ ${\displaystyle \delta _{E}(\alpha )>0}$ ${\displaystyle \alpha >0}$ ${\displaystyle \alpha \mapsto \delta _{E}(\alpha )/\alpha }$

If a sequence is in a uniformly convex space with for all and if the series is unconditionally convergent, then we have . ${\displaystyle (x_{n})_{n}}$ ${\displaystyle E}$ ${\displaystyle \|x_{n}\|\leq 2}$ ${\displaystyle n\in {\mathbb {N} }}$ ${\displaystyle \textstyle \sum _{n\in \mathbb {N} }x_{n}}$ ${\displaystyle \textstyle \sum _{n\in \mathbb {N} }\delta _{E}(\|x_{n}\|)<\infty }$

Other room classes

The uniform convexity condition discussed here is the strongest among several convexity conditions , each of which leads to different room classes. In particular, it results that evenly convex spaces are strictly convex and strongly convex and have the Radon-Riesz property .

literature

• Harro Heuser : Functional Analysis . Theory and application (=  mathematical guidelines ). 4th edition. BG Teubner, Wiesbaden 2006, ISBN 978-3-8351-0026-8 ( MR2380292 ).
• Friedrich Hirzebruch , Winfried Scharlau : Introduction to functional analysis (=  BI university pocket books . Volume 296 ). Bibliographical Institute, Mannheim / Vienna / Zurich 1971, ISBN 3-411-00296-4 ( MR0463864 ).
• Arnold Schönhage : Approximation Theory (=  de Gruyter textbook ). Walter de Gruyter & Co., Berlin, New York 1971 ( MR0277960 ).

Individual evidence

1. ↑ Friedrich Hirzebruch, Winfried Scharlau: Introduction to Functional Analysis. 1971, ISBN 3-860-25429-4 , definition 16.1
2. ↑ James A. Clarkson: Uniformly convex spaces. Transactions of the American Mathematical Society, Volume 40, 1936, pages 396-414.
3. ↑ CA McCarthy, C _p , Israel Journal of Mathematics (1967), Volume 5, pages 249-271.
4. ^ D. Milman: On some criteria for the regularity of spaces of type (B). Comptes Rendus (Doklady) de l'Académie des Sciences de l'URSS, Volume 20, 1938, pages 243-246.
5. ^ BJ Pettis: A proof that every uniformly convex space is reflexive. Duke Math. J., Volume 5, 1939, pages 249-253.
6. ↑ Mahlon M. Day: Reflexive Banach spaces not isomorphic to uniformly convex spaces. Bulletin of the American Mathematical Society, Vol. 47, No. 4, 1941, pp. 313-317.
7. ^ Arnold Schönhage: Approximation theory. 1971, p. 15
8. ^ Joseph Diestel: Sequences and Series in Banach Spaces. 1984, ISBN 0-387-90859-5 , Chapter VIII, Theorem 2