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Uniformly Convex

A uniformly convex normed space X=(X,||·||_X) is one for which, given sequences {x_n}={x_n}_(n=1)^infty, {y_n}={y_n}_(n=1)^infty, the assumptions ||x_n||_X<=1, ||y_n||_X<=1, and ||x_n+y_n||_X->2 together imply that

||x_n-y_n||_X->0

as n tends to infinity.

Such spaces are important in functional analysis. For example, the classical Banach-Saks theorem can be generalized so that the desired conclusion holds in the case that X is a Banach space whose conjugate space (that is, the dual normed space X^*) is uniformly convex.

See also

Asymptotic Midpoint Uniform Convexity, Asymptotic Uniform Convexity, Superreflexive Space

Portions of this entry contributed by Christopher Stover

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References

Rudin, W. Functional Analysis. New York: McGraw-Hill, 1991.

Referenced on Wolfram|Alpha

Uniformly Convex

Cite this as:

Weisstein, Eric W., with contributions by Christopher Stover. "Uniformly Convex." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UniformlyConvex.html

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