ANNALES POLONICI MATHEMATICI Online First version Extendability and domains of holomorphy in infinite-dimensional spaces Richard M. Aron (Kent, OH), Stéphane Charpentier (Marseille), Paul M. Gauthier (Montréal), Manuel Maestre (Valencia) and Vassili Nestoridis (Athens) Dedicated to the memory of Professor Józef Siciak Abstract. We study the notions of extendability and domain of holomorphy in the infinite-dimensional case. In this setting it is also true that the notions of domain of holomorphy and weak domain of holomorphy are equivalent. We also prove that the set of non-extendable functions belonging to some classes X(B) ⊂ H(B), B being the open unit ball in a separable complex Banach space, is a lineable and dense G δ . Moreover, when Ω is H b -holomorphically convex (defined in the text), it is shown that the set of non-extendable holomorphic functions on Ω is a lineable and dense G δ set. 1. Introduction. It is well-known (see, e.g., [20]) that the notion of domain of holomorphy and of weak domain of holomorphy are equivalent in C d ; the first proof was constructive and by no means elementary. In [13], Baire’s theorem was combined with a theorem of Banach to give a simpler proof of the above equivalence. In addition it was proved that the set of non- extendable functions contains a dense G δ set. In [17, 18] Banach’s theorem was replaced by Montel’s theorem, and with a very simple proof it was also shown that the set of non-extendable functions is itself a dense G δ set. In the latter proof it was essential that the notion of extendable function be formulated in an equivalent way so that the holomorphic extension is bounded on balls. In the proof in [17, 18], the fact that closed balls in finite dimension are compact sets was used. But in infinite dimension closed balls are no longer compact. Therefore, in order to extend the above results to 2010 Mathematics Subject Classification : Primary 46G20; Secondary 58B12. Key words and phrases : infinite-dimensional holomorphy, Baire’s theorem, Montel theo- rem, domain of holomorphy, extendability. Received 21 August 2018; revised 12 November 2018. Published online 12 April 2019. DOI: 10.4064/ap180821-5-12 [1] c  Instytut Matematyczny PAN, 2019