Convexity and Smoothness

  • Pei-Kee Lin


In the first part of this chapter (Section 2.1 and 2.2), we present some basic results about various types of convexity and smoothness conditions that the norm of a Banach space may satisfy. For convexity, we consider strictly convex, uniformly convex, locally uniformly convex, fully rotund, and nearly uniformly convex spaces. Particularly, we present a proof of the Schlumprecht-Odell theorem that shows that every separable reflexive Banach space admits an equivalent 2R norm. On the smoothness, we consider smooth, Fréchet smooth, uniformly Gâteaux smooth and uniformly smooth spaces. In the second part (Section 2.3), we introduce the spreading model, and we show that a Banach space X has the weak Banach-Saks property if and only if X does not have an ℓ1-spreading model.


Banach Space Equivalent Norm Convex Banach Space Spreading Model Smooth Point 
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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Pei-Kee Lin
    • 1
  1. 1.Department of MathematicsUniversity of MemphisMemphisUSA

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