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
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
- 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
- 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
References
Citations
General references
- Clarkson, J. A. (1936). "Uniformly convex spaces". Trans. Amer. Math. Soc. 40 (3). American Mathematical Society: 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.