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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. §1. Notation and PreliminariesIn the study of the space of holomorphic functions on infinite dimensional complex normed spaces H(E), several natural locally convex topologies appear. Among them, the three most important are the compact open topology τ 0 , the Nachbin "ported" topology τ ω , and the bornological topology τ δ associated to these two topologies. One useful characterization of the τ δ topology, introduced independently by Nachbin [6] and Coeuré [2] about 30 years ago, is that it is generated by all seminorms p : H(E) → C which satisfy the following property:

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