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  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 . The existence of a basis for the full tensor product was proved by Gelbaum and Gil de Lamadrid  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  who treated this issue in the context of spaces of matrices and by Pisier  and Schütt . The dual problem, whether the monomials are a basis in the space of homogeneous polynomials, was dealt with by Dimant in her thesis , as well as in two other articles, together with Dineen  and Zalduendo . The unconditionality (or lack thereof) of the monomial basis was extensively analysed by Defant, Díaz, García and Maestre .