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 $1<p<\infty$. 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 $\|x\|\leq 1$ and $\|y\|\leq 1$, $\|x+y\|>2-\delta$ implies $\|x-y\|<\epsilon$.

Properties

Every uniformly convex space is reflexive (see Ringrose (1959)).

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.
• J. R. Ringrose, A note on uniformly convex spaces, J. London Math. Soc. 34 (1959), 92.