Uniformly convex space

Uniformly convex space

In mathematics, uniformly convex spaces are common examples of reflexive Banach spaces. These include all Hilbert spaces and the L"p" spaces for 1 The concept of uniform convexity was first introduced by James A. Clarkson in 1936.

Definition

A uniformly convex space is a Banach space so that, for every epsilon>0 there is some delta>0 so that for any two vectors with |x|le1 and |y|le 1,

:|x+y|>2-delta

implies

:|x-y|

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 space is reflexive.

ee also

* Modulus and characteristic of convexity
* Hanner's inequalities say that L"p" spaces (1 are uniformly convex.

References

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