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.

Basic topics on Banach space theory needed for the text are reviewed. Hahn-Banach theorem, Baire’s theorem, uniform boundedness principle, closed graph theorem, weak topologies, Banach-Alaoglu theorem, unconditional basis, Banach sequence spaces, summing operators, factorable operators, cotype, Kahane inequality.

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.

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.