Weak moduly of convexity

• Autores: Antonio Ullán de Celis, Javier Alonso Romero
• Localización: Extracta mathematicae, ISSN-e 0213-8743, Vol. 6, Nº 1, 1991, págs. 47-49
• Idioma: español
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• Resumen

  • Let E be a real normed linear space with unit ball B and unit sphere S. The classical modulus of convexity of J. A. Clarkson [2] dE(e) = inf {1 - 1/2||x + y||: x,y Î B, ||x - y|| = e} (0 = e = 2) is well known and it is at the origin of a great number of moduli defined by several authors. Among them, D. F. Cudia [3] defined the directional, weak and directional weak modulus of convexity of E, respectively, as dE(e,g) = inf {1 - 1/2||x + y||: x,y Î B, g(x-y) = e} dE(e,f) = inf {1 - 1/2 f(x,y): x,y Î B, ||x - y|| = e} dE(e,f,g) = inf {1 - 1/2 f(x,y): x,y Î B, g(x-y) = e} where 0 = e = 2 and f,g Î S' (unit sphere of the topological dual space E').

    D. F. Cudia [3] has shown the close connection existing between these moduli and various differentiability conditions of the norm in E'.

    In this note we study these moduli from a different point of view, then we analyze some of its properties and we see that it is possible to characterize inner product spaces by means of them.