Schauder Bases for Symmetric Tensor Products

  • Bogdan C. Grecu

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

    National University of Ireland, Galway, Ireland


For a Banach space with Schauder basis, we prove that the fold symmetric tensor product has a Schauder basis for all symmetric uniform crossnorms . This is done by modifying the square ordering on and showing that the new ordering gives tensor product bases in both and .

The main purpose of this article is to prove that the -fold symmetric tensor product of a (real or complex) Banach space has a Schauder basis whenever 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 , 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 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].

Cite this article

Bogdan C. Grecu, Raymond A. Ryan, Schauder Bases for Symmetric Tensor Products. Publ. Res. Inst. Math. Sci. 41 (2005), no. 2, pp. 459–469

DOI 10.2977/PRIMS/1145475364