Schauder Bases for Symmetric Tensor Products

Abstract

For a Banach space $E$ with Schauder basis, we prove that the $n$ fold symmetric tensor product ${\hat{\otimes }}_{\mu ,s}^{n}E$ has a Schauder basis for all symmetric uniform crossnorms $\mu$. This is done by modifying the square ordering on $N^n$ and showing that the new ordering gives tensor product bases in both ${\hat{\otimes }}_{\mu }^{n}E$ and ${\hat{\otimes }}_{\mu ,s}^{n}E$.

The main purpose of this article is to prove that the $n$-fold symmetric tensor product of a (real or complex) Banach space $E$ has a Schauder basis whenever $E$ does. The result was stated without proof in Ryan’s thesis [11] and has been constantly referred to in the literature. In the particular case of a shrinking Schauder basis for a complex Banach space $E$, an implicit proof was given by Dimant and Dineen [2]. The existence of a basis for the full tensor product was proved by Gelbaum and Gil de Lamadrid [7] who also showed that the unconditionality of the basis for $E$ does not necessarily imply the same property for the tensor product basis. This was taken further by Kwapień and Pełczyński [8] who treated this issue in the context of spaces of matrices and by Pisier [9] and Schütt [12]. The dual problem, whether the monomials are a basis in the space of homogeneous polynomials, was dealt with by Dimant in her thesis [1], as well as in two other articles, together with Dineen [2] and Zalduendo [3]. The unconditionality (or lack thereof) of the monomial basis was extensively analysed by Defant, Díaz, García and Maestre [4].