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.

Given a Banach space X, we consider Hardy spaces of X-valued functions on the infinite polytorus, Hardy spaces of X-valued Dirichlet series (defined as the image of the previous ones by the Bohr transform), and Hardy spaces of X-valued holomorphic functions on l_2 ∩ B_{c0}. The chapter is dedicated to study the interplay between these spaces. It is shown that the space of functions on the polytorus always forms a subspace of the one of holomorphic functions, and these two are isometrically isomorphic if and only if X has ARNP. Then the question arises of what do we find in the side of Dirichlet series when we look at the image of the Hardy space of holomorphic functions. This is also answered, showing that this consists of Dirichlet series for which all horizontal translations (those whose coefficients are (a_n/n^ε)) are in \mathcal{H}_p with uniformly bounded norms. Also, a version of the brothers Riesz theorem for vector-valued functions is given.

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.