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Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence spaces

M. Jimenéz Sevilla;  Rafael Payá — 1998

Studia Mathematica

For each natural number N, we give an example of a Banach space X such that the set of norm attaining N-linear forms is dense in the space of all continuous N-linear forms on X, but there are continuous (N+1)-linear forms on X which cannot be approximated by norm attaining (N+1)-linear forms. Actually,X is the canonical predual of a suitable Lorentz sequence space. We also get the analogous result for homogeneous polynomials.

Convex sets in Banach spaces and a problem of Rolewicz

A. Granero;  M. Jiménez Sevilla;  J. Moreno — 1998

Studia Mathematica

Let B x be the set of all closed, convex and bounded subsets of a Banach space X equipped with the Hausdorff metric. In the first part of this work we study the density character of B x and investigate its connections with the geometry of the space, in particular with a property shared by the spaces of Shelah and Kunen. In the second part we are concerned with the problem of Rolewicz, namely the existence of support sets, for the case of spaces C(K).

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