Uniformly convex space

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

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A uniformly convex space is a normed vector space such that, for every   there is some   such 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

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  • 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 that 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

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  • Every inner-product space is uniformly convex.[1]
  • Every closed subspace of a uniformly convex Banach space is uniformly convex.
  • Clarkson's inequalities imply that Lp spaces   are uniformly convex.
  • Conversely,   is not uniformly convex.

See also

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References

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Citations

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  1. ^ Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces (2nd ed.). Boca Raton, FL: CRC Press. p. 524, Example 16.2.3. ISBN 978-1-58488-866-6.

General references

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