Uniformly smooth space

In mathematics, a uniformly smooth space is a normed vector space $X$ satisfying the property that for every $\epsilon >0$ there exists $\delta >0$ such that if $x,y\in X$ with $\|x\|=1$ and $\|y\|\leq \delta$ then

$\|x+y\|+\|x-y\|<2+\epsilon \|y\|$.

Properties

• Every uniformly smooth Banach space is reflexive.
• A Banach space $X$ is uniformly smooth if and only if its dual $X^{*}$ is uniformly convex (and vice versa, via reflexivity).
• A Banach space is uniformly smooth if and only if the limit

$\lim _{t\to 0}{\frac {\|x+ty\|-\|x\|}{t}}$

exists uniformly for all $x,y\in S_{X}$ (where $S_{X}$ denotes the unit sphere of $X$).

See also

• Uniformly convex space

References

• Diestel, Joseph (1984). Sequences and Series in Banach spaces. Springer-Verlag New York Berlin Heidelberg Tokyo. ISBN 0-387-90859-5.

• Itō, Kiyoshi (1993). Encyclopedic Dictionary of Mathematics, Volume 1. MIT Press. ISBN 0-262-59020-4. [1]