Plurisubharmonic domination in Banach spaces

We show that if X is a Banach space with a Schauder basis, Ω⊂X is a pseudoconvex open subset, and u:Ω→(−∞,∞) is a locally bounded function, then there is a continuous plurisubharmonic function w:Ω→(−∞,∞) with u(x)⩽w(x) for all x∈Ω. This has many applications to analytic cohomology of complex Banach manifolds.