The set of monomial convergence of the bounded holomophic functions on B_{c0} and of m-homogeneous polynomials on c0 was studied in Chapter 10. Here the space c0 is replaced by some other l_p spaces, or even by polynomials on an arbitrary Banach sequence space and holomorphic functions on Reinhardt domains. The only complete case is p=1, where the set of monomial convergence of the m-homogeneous polynomials is exactly l_1, and the set of monomial convergence of the bounded holomorphic functions on the open unit ball of l_1 is again the ball. For other p’s upper and lower bounds are presented that give a pretty tight description.

Given a family of formal power series, its set of monomial convergence is defined as those z’s for which the series converges. The main focus is given to the sets of monomial convergence of the m-homogeneous polynomials on c0 and of the bounded holomorphic functions on B_{c0}. The first one is completely described in terms of the Marcinkiewicz space l_{(2m)/(m-1), ∞}. For the second one there is no complete description. If z is such that limsup (log n)^(1/2) ∑_j^n (z*_j)^{2} < 1 (where z* is the decreasing rearrangement of z), then z is in the set of monomial convergence of the bounded holomorphic functions. Also, if z belongs to the set of monomial convergence, then the limit superior is ≤ 1. This is related to Bohr’s problem (see Chapter 1). First of all, if M denotes the supremum over all q so that l_q is contained in the set of monomial convergence of the bounded holomorphic functions on Bc0, then S=1/M. But this can be more precise: S is the infimum over all σ >0 so that the sequence (p_n^(-σ))_n (being p_n the n-th prime number) belongs to the set of monomial convergence of the bounded holomorphic functions on Bc0.

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.

Each Hardy space of Dirichlet series \mathcal{H}_p has an associated abscissa, and the analogue to Bohr’s problem arises in a natural way: to determine the maximal distance S_p between this abscissa and the abscissa of absolute convergence. If a Dirichlet series with coefficients (a_n) belongs to \mathcal{H}_p, then the series with coefficients (a_n/n^{ε}) belongs to \mathcal{H}_q for all q>p and ε >0. It is shown that S_p=1/2, and that, if we only consider m-homogeneous Dirichlet series, S_p^m=1/2. For every 1 ≤ p < ∞ the set of monomial convergence of the Hardy space H_p of functions on the infinite dimensional polytorus (hence also of the Hardy space H_2 on the infinite-dimensional polytorus) is l_2 ∩ Bc0. The space of all multipliers on the Hardy space of Dirichlet series \mathcal{H}_p coincides with \mathcal{H}_\infty.

For each 1 ≤ p ≤ ∞, the Hardy space \mathcal{H}_p of Dirichlet series is defined as the image through the Bohr transform of the Hardy space of functions on the infinite-dimensional polytorus. The Dirichlet polynomials are dense in \mathcal{H}_p for every 1 ≤ p < ∞. For p=2 this coincides with the space of Dirichlet series whose coefficients are square-summable. A Dirichlet series with coefficients a_n belongs to\mathcal{H}_p if and only if the series with coefficients a_n/n^ε is in \mathcal{H}_p for every ε >0 and the norms are uniformly bounded. In this case, the series is the limit as ε tends to 0. As a technical tool to see this, vector-valued Dirichlet series (that is, series with coefficients in some Banach space) are introduced, and some basic definitions and properties (such as the convergence abscissas, Bohr-Cahen formulas) are given.