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The generalized Rademacher functions. (English) Zbl 0844.46002
In [R. M. Aron and J. Globevnik, Rev. Mat. Univ. Complutense, Madr. 2, 27-34 (1989; Zbl 0748.46021)], the authors introduced the so-called generalized Rademacher functions and used them to prove that every continuous multilinear form \(A: c_0 \times \dots \times c_0\to \mathbb{C}\) has a trace. In this note, we show that these functions are quite useful in obtaining simple proofs of various estimates in several different areas of analysis. We first give a simple proof of a result of I. Zalduendo [Proc. R. Ir. Acad., Sect. A 93, No. 1, 137-142 (1993; Zbl 0790.46016)], which extends the result cited above from \(c_0\) to \(\ell_p\). Next, we show how generalized Rademacher functions can be used to provide a new proof of a theorem of A. Defant and J. Voigt [see, for example, R. Alencar and M. Matos, Some classes of multilinear mappings between Banach spaces, Preprint, Univ. Complutense de Madrid 12 (1989)]. Then we exhibit this theorem as a special case of a more general result, which in turn yields other consequences, one new and one old. Next, we use these functions to derive a new polarization formula for symmetric multilinear forms, which yields a new proof of an inequality of Harris. Finally, we provide a simple proof of a theorem of Pełczynski about the continuity of multilinear mappings with respect to a certain sequential topology.

46A32 Spaces of linear operators; topological tensor products; approximation properties
46A45 Sequence spaces (including Köthe sequence spaces)
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)