Let F be either the set of all bounded holomorphic functions or the set of all m-homogeneous polynomials on the unit ball of ℓ r . We give a systematic study of the sets of all u ∈ ℓ r for which the monomial expansion α ∂ α f (0) α! u α of every f ∈ F converges. Inspired by recent results from the general theory of Dirichlet series, we establish as our main tool, independently interesting, upper estimates for the unconditional basis constants of spaces of polynomials on ℓ r spanned by finite sets of monomials.We use standard notation from Banach space theory. As usual, we denote the conjugate exponent of 1 ≤ r ≤ ∞ by r ′ , i.e. 1 r + 1 r ′ = 1. Given m, n ∈ N we consider the following sets of indices M (m, n) = j = ( j 1 , . . . , j m ) ; 1 ≤ j 1 , . . . , j m ≤ n = {1, . . . , n} mFor indices i, j ∈ M we denote by (i, j) = (i 1 , i 2 , . . . , j 1 , j 2 , . . . ) the concatenation of i and j. An equivalence relation is defined in M (m) as follows: i ∼ j if there isGiven a Banach sequence space X and some index subset J ⊂ J , we write P ( J X ) for the closed subspace of all holomorphic functionswhere z j for j = ( j 1 , . . . , j ℓ ) stands for the monomial z j : u → u j := u j 1 ·. . . ·u j ℓ . For J ⊂ J (m), we call J * = j ∈ J (m − 1); ∃k ≥ 1, (j, k) ∈ J the reduced set of J .