JournalsprimsVol. 41 , No. 2DOI 10.2977/prims/1145475364

Schauder Bases for Symmetric Tensor Products

  • Bogdan C. Grecu

    National University of Ireland, Galway, Ireland
  • Raymond A. Ryan

    National University of Ireland, Galway, Ireland
Schauder Bases for Symmetric Tensor Products cover


For a Banach space E with Schauder basis, we prove that the n fold symmetric ⊗ˆµ_n__E_ has a Schauder basis for all symmetric uniform crossnorms µ. This is done by modifying the square ordering on ℕ_n_ and showing that the new ordering gives tensor product bases in both ⊗ˆµ_n__E_ and ⊗ˆµ,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 Pelczyński [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].