Uniformly convex space
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In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936.
A uniformly convex space is a normed vector space so that, for every there is some so that for any two vectors with and the condition
Intuitively, the center of a line segment inside the unit ball must lie deep inside the unit ball unless the segment is short.
- The Milman–Pettis theorem states that every uniformly convex Banach space is reflexive, while the converse is not true.
- If is a sequence in a uniformly convex Banach space which converges weakly to and satisfies then converges strongly to , that is, .
- A Banach space is uniformly convex if and only if its dual is uniformly smooth.
- Every uniformly convex space is strictly convex.
- Every Hilbert space is uniformly convex.
- Every closed subspace of a uniformly convex Banach space is uniformly convex.
- Hanner's inequalities imply that Lp spaces are uniformly convex.
- Conversely, is not uniformly convex. For example, in consider and . Then and , but .
- Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.