Uniformly convex space

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

Motivation and definition

Since the ball unit of a normalized area is convex, the center of which lies between two vectors and the unit sphere back into the unit sphere. We investigate the distance of such a center from the edge of the unit sphere.

Referring to the Euclidean norm, the unit ball, the unit circle in the plane. If one forms the center of two boundary points, so this focus is more on the inside of the circle, the more the two edge points are from each other.

Thus, it is a particular geometric feature that two vectors of the ball unit to be close to each other when the center of which is located close to the edge. Therefore, we define:

A normed space is called uniformly convex if, for every a, so that the following holds: If with, and, as follows.

This is a feature of the standard. Going to an equivalent norm on, so this property can be lost, as the two above- considered examples.

Examples

  • One can easily show by means of the parallelogram that inner product spaces are uniformly convex.
  • JA Clarkson has this property for the Banach spaces Lp [ 0,1], demonstrated ( set of Clarkson ). This statement has been considerably generalized in 1950 by EJ McShane. Is a uniformly convex space, any positive measure, so is also uniformly convex. Here is the Banach space of equivalence classes of measurable functions with values ​​in so.

Set of Milman

David Milman has proved another important property uniformly convex spaces:

Set of Milman: Uniformly convex Banach spaces are reflexive.

This result has been found independently by Milman also by Billy James Pettis (1913-1979), which is why it is sometimes called the set of Milman - Pettis. The class of uniformly convex spaces is strictly smaller than the class of reflexive spaces, because there are reflexive Banach spaces which are not isomorphic to uniformly convex spaces.

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

Strict convexity

A normed space is called strictly convex (or strictly normalized) if and always follows. One can easily show that uniformly convex spaces are strictly convex. One therefore has the following inclusions of classes of normed spaces:

Inner product spaces Uniformly convex spaces strictly convex spaces Normed spaces.

The approximation theorem

The following statements, which are also known as approximation theorem, show the importance of uniformly convex spaces for the approximation theory. Many approximation problems can be reformulated so that in a convex set (eg in a subspace ) is to find a vector to a given vector has the shortest distance. There are the following statements for a real normed space, and a closed and convex subset:

  • Uniqueness: Is strictly convex, then there is at most one with.
  • Existence: Is a uniformly convex Banach space, then there is a ( uniquely determined by the above ) with.

Convexity

It sets a number

And calls the function defined by the convexity of. For uniformly convex spaces is true by definition for all, and one can show that the convexity is a monotonic function, even the picture is monotonic. A set of M. Kadets is a necessary condition for the unconditional convergence of series in uniformly convex spaces is:

If a sequence in a uniformly convex space with for all and is the series absolutely convergent, so true

Schatten class operator Shizuo Kakutani
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