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 Bohnenblust-Hille inequality bounds the (2m)/(m+1)-norm of the coefficients of an m-homogeneous polynomial in n variables by a constant (depending on m but not on n) multiplied by the norm (the supremum on the n-dimensional polydisc) of the polynomial. This follows from the inequality for m-linear forms. Littlewood’s inequality shows that the 4/3-norm of a bilinear form is bounded by a constant (not depending on n) multiplied by the norm of the form and that 4/3 cannot be improved. A tool is the Khinchin-Steinhaus inequality, showing that the L_p-norms (for 1 ≤ p < ∞) of a polynomial are equivalent to the l_2 norm of the coefficients. Other tools are inequalities relating mixed norms of the coefficients of a matrix with the norm of the associated multilinear form. All these give the multilinear Bohnenblust-Hille inequality, showing also that the (2m)/(m+1) cannot be improved. The exponent in the polynomial inequality is also optimal (this does not follow from the multilinear case). As a consequence of the inequality we have S^m=(2m)/(m-1) (see Chapter 4). By a generalized Hölder inequality the constant in the multilinear inequality grows at most polynomially on m.