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 work with integrable functions on the polytorus, both in finite and infinitely many variables. For such a function and a multi-index the corresponding Fourier coefficient is defined. For each 1 ≤ p ≤ ∞ the Hardy space H_p consists of those functions in L_p having non-zero Fourier coefficients only for multi-indices in the positive cone. The Hardy space H_\infty on the infinite dimensional polytorus and the space of bounded holomorphic functions on Bc0 are isometrically isomorphic. To prove this the Poisson kernel in several variables is defined, and the Poisson operator (defined through convolution with this kernel) is considered. With these it is shown that the trigonometric polynomials are dense in L_p for 1 ≤ p < ∞ and weak*-dense in L_\infty, and that so also are the analytic trigonometric polynomials in H_p and H_∞. The isometry between the two spaces is first established for the finite dimensional polytorus/polydisc and then, using a version of Hilbert’s criterion (see Chapter 2), raised to the infinite-dimensional case. The density of the polynomials can be proved using the Féjer kernel instead of the Poisson one.

We establish a bijection between Dirichlet series and formal power series through Bohr’s transform. This is one of the main tools all along the text and relies on the fact that by the fundamental theorem of arithmetic every natural number has a unique decomposition as a product of prime numbers. In this way, to each such number a multi-index can be assigned (and vice-versa). With this we show that the space of bounded holomorphic functions on B_{c0} and \mathcal{H}_\infty are isomorphic as Banach spaces. This means that to every holomorphic function corresponds a Dirichlet series in such a way that the monomial and the Dirichlet coefficients are identified. We consider m-homogenous Dirichlet series: those having non-zero coefficients only if n has exactly m prime divisors (counted with multiplicity) and show that the space of such Dirichlet series is isometrically isomorphic to the space of m-homogeneous polynomials on c0.

The Bohr radius for p-norms was introduced and studied in Chapter 19. There it was shown that unconditional basis constants of the monomials in spaces of m-homogeneous polynomials and Bohr radii are, in a certain sense, reciprocal to each other. In Chapter 21 the Gordon-Lewis cycle of ideas was developed to study these unconditional basis constants. Relating unconditional basis constants, Gordon-Lewis constants and projection constants of spaces of m-homogeneous polynomials gives a new proof of the lower bound for the Bohr radius for p-norms.

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.

We establish the basic notions around Dirichlet series that are going to be used all along the text. A Dirichlet series converges on half-planes, and that there it defines a holomorphic function. For a given Dirichlet series we consider four abscissas definining the maximal half-planes on which it: converges, defines a bounded holomorphic function, converges uniformly or converges absolutely. We formulate the problem of determining the maximal possible distance between these abscissas. The difference between the abscissa of convergence and absolute convergence is at most one, and this is attained. Also, the abscissa of uniform convergence and of boundedness always coincide (this is Bohr theorem). Then Bohr’s problem is established: to determine S, the maximal possible width of the strip of absolute but not uniform convergence of Dirichlet series, and we show that it is at most 1/2. Finally we introduce the Banach space \mathcal{H}_\infty of Dirichlet series that converge and define a bounded holomorphic function on the right half-plane and reformulate Bohr’s problem in terms of this space. This becomes later an important tool.

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.

We give an alternative, probabilistic, approach to two of the subjects considered so far: the optimality of the exponent in the polynomial Bohnenblust-Hille inequality (see Chapter 6) and the lower bound for S in Bohr’s problem (see Chapters 1 and 4). We use a probabilistic device: the Kahane-Salem-Zygmund inequality. This shows that, for a given finite family of coefficients, a choice of signs can be found in such a way that the polynomial whose coefficients are the original ones multiplied by the signs has small norm (supremum on the polydisc). The proof uses Bernstein’s inequality and Rademacher random variables. We also look at the relationship between Rademacher and Steinhaus random variables, and deduce the classical Khinchin inequality from the Khinchin-Steinhaus inequality (see Chapter 6). We consider Dirichlet series, place signs before the coefficients, and define the almost sure abscissas (in each of the senses from Chapter 1) by considering each convergence for almost every choice of signs. An analogue of Bohr’s problem in this sense is considered.

Given a function f on the n-dimensional polydisc, the Bohr radius (recall Chapter 8) looks for the best r for which the supremum of ∑ | c_α z^α| for || z ||_∞ <r is less than or equal to the supremum of |f(z)| for || z ||_∞ <1. Here an analogous problem is considered, replacing the sup-norm by another p-norm. The corresponding Bohr radius for l_p-balls is defined, and its asymptotic behaviour is computed. This is done in three steps. First, an m-homogeneous version (where only m-homogeneous polynomials are considered) is defined, and it is shown how these m-homogeneous radii determine the general Bohr radius. In the second step, this homogenous radius is related to the unconditional basis constant of the monomials in the space of homogeneous polynomials on l_p. Finally, this unconditional basis constant is computed.

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.

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.

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.