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.
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. Intuitively, the strict convexity means a stronger triangle inequality whenever are linearly independent, while the uniform convexity requires this inequality to be true uniformly.
- 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.
- 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) . 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.