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We prove, in a general framework, the existence of a closed infinitedimensional subspace consisting of universal series. Introduction. The study of universal series began in 1914, when Fekete showed that there exists a Taylor series on [−1, 1] whose subsequences of partial sums approximate uniformly any continuous function f on [−1, 1] such that f(0) = 0. The main idea of this result is that whatever can be uniformly approximated by polynomials can also be approximated by subsequences of partial sums of a Taylor series. After this, a lot of examples of universal series appeared in different spaces: Men’shov showed, for instance, the existence of many trigonometric series which approximate almost everywhere, along subsequences of their partial sums, any complex measurable 2π-periodic function on the torus. Other pointwise universality results were obtained concerning Faber series, Laurent series, Jacobi series, harmonic expansions, etc. For such results, we refer to the two surveys [6] and [7]. It turns out that, when a universal series does exist, the set of universal series has some nice properties. This was developed in [5] (in a more general context) and later in [2] and [14], where an abstract theory of universal series is presented. Precisely, it is shown there that, as soon as there exists a universal series, then the set of universal series is both generic (it contains a dense Gδ set) and algebraically generic (it contains a dense subspace, without 0). This is done in a general framework which covers every known example. Independently, Bayart studied in [1] the existence of a closed infinitedimensional subspace of universal series. His work just deals with one particular example, that of universal Taylor series of holomorphic functions on the unit disk D. Our aim, in this paper, is to combine the ideas of [1] and [2] 2010 Mathematics Subject Classification: 30Bxx, 30B30.

On the closed subspaces of universal series in Banach spaces and Fréchet spaces