We prove the impossibility of expressing the space of entire functions on any infinite dimensional complex Banach space with a Schauder basis E as a countable union of spaces of entire functions that are bounded on countable open covers of E.
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José M. Ansemil, Richard M. Aron, Socorro Ponte, Representation of Spaces of Entire Functions on Banach Spaces. Publ. Res. Inst. Math. Sci. 45 (2009), no. 2, pp. 383–391