Monomial bases for holomorphic functions on Banach spaces with an unconditional basis
Let $X$ be a complex Banach space with an unconditional Schauder basis. We prove that the monomials associated with this basis, endowed in each homogeneous degree with the square order and globally with any compatible ordering, form a Schauder basis for the space $(\calH(X),\tau_0)$ of entire holomorphic functions on $X$ with the compact-open topology. The proof relies on two main ideas. First, coordinate-tail conditions yield a fundamental system of compact solid subsets of $X$. Second, Fourier projections in the coordinates give the uniform estimate $c_{K,n}\leq n+1$ for the basis constant of the degree-$n$ monomials with respect to the supremum seminorm on each such compact set $K$. We derive applications to Banach sequence lattices with dense $c_{00}$, including classical, Lorentz, Orlicz-heart and Schreier-type sequence spaces, and to general vector-valued $E$-sums. In particular, the result covers mixed $c_0$- and $\ell_r$-sums of finite-dimensional $\ell_p$ spaces and the Lorentz predual $d_*(w,1)$.