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.

This is a short introduction to the theory of holomorphic functions in finitely and infinitely many variables. We begin with functions in finitely many variables, giving the definition of holomorphic function. Every such function has a monomial series expansion, where the coefficients are given by a Cauchy integral formula. Then we move to infinitely many variables, considering functions defined on B_{c0}, the open unit ball of the space of null sequences. Holomorphic functions are defined by means of Fréchet differentiability. We have versions of Weierstrass and Montel theorems in this setting. Every holomorphic function on B_{c0} defines a family of coefficients through a Cauchy integral formula and a (formal) monomial series expansion. Every bounded analytic (represented by a convergent power series) function is holomorphic. Hilbert’s criterion, that gives conditions on a family of scalars so that it is attached to a bounded holomorphic function on B_{c0}. Homogeneous polynomials are those entire functions having non-zero coefficients only for multi-indices of a given order. We show how these are related to multilinear forms on c0 through the polarization formulas.

We give the solution of Bohr’s problem, showing that in fact S=1/2. This is done by considering an analogous problem where only m-homogeneous Dirichlet series are taken into account (defining, then, S^m). Using the isometry between homogeneous Dirichlet series and polynomials, the problem is translated into a problem for these. For each m we produce an m-homogeneous polynomial P such that for every q > (2m)/(m-1) there is a point z in l_q for which the monomial series expansion of P does not converge at z. This shows that, contrary to what happens for finitely many variables, holomorphic functions in infinitely many variables may not be analytic. This also shows that (2m)/(m-1) ≤ S^m for every m and then gives the result. There is more. For each fixed 0 ≤ σ ≤ 1/2 there is a Dirichlet series whose abscissas of uniform and absolute convergence are at distance exactly σ.

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 give an introduction to vector-valued holomorphic functions in Banach spaces, defined through Fréchet differentiability. Every function defined on a Reinhardt domain of a finite-dimensional Banach space is analytic, i.e. can be represented by a monomial series expansion, where the family of coefficients is given through a Cauchy integral formula. Every separate holomorphic (holomorphic on each variable) function is holomorphic. This is Hartogs’ theorem, which is proved using Leja’s polynomial lemma. For infinite-dimensional spaces, homogeneous polynomials are defined as the diagonal of multilinear mappings. A function is holomorphic if and only if it is Gâteaux holomorphic and continuous, if and only if it has representation as a series of homogeneous polynomials (known as Taylor expansion). A function is weak holomorphic if the composition with every functional is holomorphic. A function is holomorphic if and only if it is weak holomorphic. Analytic functions are holomorphic.