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Let E be an infinite dimensional complex Banach space. We prove the existence of an infinitely generated algebra, an infinite dimensional closed subspace and a dense subspace of entire functions on E whose non-zero elements are functions of unbounded type. We also show that the τδ topology on the space of all holomorphic functions cannot be obtained as a countable inductive limit of Fr´echet spaces. RESUMEN. Sea E un espacio de Banach complejo de dimension infinita y sea H(E) el espacio de funciones holomorfas definidas en E. En el articulo se demuestra la existencia de un algebra infinitamente generada en H(E), un subespacio vectorial en H(E) cerrado de dimension infinita y un subespacio denso en H(E) cuyos elementos no nulos son funciones de tipo no acotado. Tambien se demuestra que el espacio de funciones holomorfas con la topologia ? no es un limite inductivo numberable de espacios de Frechet.

Vector spaces of entire functions of unbounded type