Modulus of convexity in Banach spaces

✍ Scribed by J Gao

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 381 KB
• Volume: 16
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/s0893-9659(03)80043-5

✦ Synopsis

and 0 < e < 2 is the modulus of convexity of X. The best results so far about the relationship between normal structure and the modulus of convexity of X are that for any Banach space X either 6(1) > 0 or 6(3/2) > 1/4 implies X has normal structure. We generalize the above results in this paper to prove that for any Banach space X, 5(1 + e) > e/2 for any e, 0 < e < 1, implies X has uniform normal structure. (~) 2003 Elsevier Science Ltd. All rights reserved.

Keywords--Arc length, Modulus of convexity, Normal structure, Uniformly nonsquare space, Uniform normal structure and ultraproduct space.

Let X be a normed linear space, and let S(X)= {x • X: Ilxll = 1} and B(X)= {x • X : Itxll < 1} be the unit sphere and the unit ball of X, respectively.