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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.

##### MSC:
 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.)