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 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. Intuitively, the strict convexity means a stronger triangle inequality whenever are linearly independent, while the uniform convexity askes for 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.