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.

Definition[edit]

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

implies that:

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

Properties[edit]

  • The unit sphere can be replaced with the closed unit ball in the definition. Namely, a normed vector space is uniformly convex if and only if for every there is some so that, for any two vectors and in the closed unit ball (i.e. and ) with , one has (note that, given , the corresponding value of could be smaller than the one provided by the original weaker definition).
Proof

The "if" part is trivial. Conversely, assume now that is uniformly convex and that are as in the statement, for some fixed . Let be the value of corresponding to in the definition of uniform convexity. We will show that , with .

If then and the claim is proved. A similar argument applies for the case , so we can assume that . In this case, since , both vectors are nonzero, so we can let and . We have and similarly , so and belong to the unit sphere and have distance . Hence, by our choice of , we have . It follows that and the claim is proved.

  • The Milman–Pettis theorem states that every uniformly convex Banach space is reflexive, while the converse is not true.
  • Every uniformly convex Banach space is a Radon-Riesz space, that is, 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. Intuitively, the strict convexity means a stronger triangle inequality whenever are linearly independent, while the uniform convexity requires this inequality to be true uniformly.

Examples[edit]

  • 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.

See also[edit]

References[edit]

  • Clarkson, J. A. (1936). "Uniformly convex spaces". Trans. Amer. Math. Soc. American Mathematical Society. 40 (3): 396–414. doi:10.2307/1989630. JSTOR 1989630..
  • Hanner, O. (1956). "On the uniform convexity of and ". Ark. Mat. 3: 239–244. doi:10.1007/BF02589410..
  • Beauzamy, Bernard (1985) [1982]. Introduction to Banach Spaces and their Geometry (Second revised ed.). North-Holland. ISBN 0-444-86416-4.
  • Per Enflo (1972). "Banach spaces which can be given an equivalent uniformly convex norm". Israel Journal of Mathematics. 13 (3–4): 281–288. doi:10.1007/BF02762802.
  • Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.