Uniformly convex space

From Wikipedia, the free encyclopedia

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

[edit] Definition

A uniformly convex space is a Banach space so that, for every ε > 0 there is some δ > 0 so that for any two vectors with and implies

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

[edit] Properties

The Milman–Pettis theorem states that every uniformly convex space is reflexive.

[edit] References

• J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396–414.
• O. Hanner, On the uniform convexity of L^p and l^p, Ark. Mat. 3 (1956), 239–244.