The text is closed by coming back to Bohr’s absolute convergence problem, this time for vector-valued Dirichlet series. For a Banach space X abscissas and strips S(X) and S_p(X), analogous to those defined in Chapters 1 and 12 are considered. It is shown that all these strips equal 1-1/cot(X), where cot(X) is the optimal cotype of X.

Basic topics on Banach space theory needed for the text are reviewed. Hahn-Banach theorem, Baire’s theorem, uniform boundedness principle, closed graph theorem, weak topologies, Banach-Alaoglu theorem, unconditional basis, Banach sequence spaces, summing operators, factorable operators, cotype, Kahane inequality.

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.

We look for inequalities that relate some p-norm of the coefficients of a vector-valued polynomial in n variables with a constant (that depends on the degree but not on n) and the supremum of the polynomial on the n-dimensional polydisc (or other n-dimensional balls) . This is an analogue of the Bohnenblust-Hille (and the Hardy-Littlewood inequalities) for vector-valued polynomials that have been extensively studied. This leads in a natural way to cotype. It is shown that if the polynomial takes values in a Banach space with cotype q, then such an inequality is satisfied with the q-norm of the coefficients. The constant that appears grows too fast on the degree. If we want to have a better asymptotic behaviour of the constants a finer property on the space is needed: hypercontractive polynomial cotype. Conditions are given for a space to enjoy this property. A polynomial version of the Kahane inequality is given (all L_p norms are equivalent for polynomials). Finally, these type of inequalities is extended to operators between Banach spaces, leading to the definition of polynomially summing operators, an extension of the classical concept of summing operator.