21 - Dineen’s Problem

Summary

The following question is addressed: if X is an infinite dimensional Banach space with unconditional basis, does the space of m-homogeneous polynomials have an unconditional basis for m ≥ 2? The purpose of this chapter is to show that the answer is negative. The proof is done in three steps. The first step shows that X contains the l_2^n’s or the l_\infty^n’s uniformly complemented whenever the space of m-homogeneous is separable. The second step proves that, if the space of m-homogeneous polynomials on X has an unconditional basis, then the unconditional basis constants of the monomials in the spaces of m-homogeneous polynomials on l_2^n and l_\infty^n are bounded (in n). But the third step shows that these unconditional basis constants in fact are not bounded. The first step uses greedy bases and spreading models. The second step goes through a cycle of ideas developed by Gordon and Lewis, relating the unconditional basis constant of a space to its Gordon-Lewis constant. The third step is given with the probabilistic devices developed in Chapter 17.