Holomorphic functions taking values in quotients of Fréchet \(b\)-spaces (Q2479744)

Let \(U\) be an open connected subset of \(\mathbb{C}^N\) and \(E| F\) be a quotient bornological space. The space of holomorphic functions from \(U\) to \(E| F\) is defined as \(O_1(U, E| F) = \text{ proj}_V O(V,E| F)\), where \(O(V,E| F) \simeq O(V,E)| O(V,F)\) and \(V\) runs through the relatively compact open subsets of \(U\). The main result of the present paper is: If \(E\) and \(F\) are Fréchet b-spaces, then \(O_1(U, E| F) \simeq O(U)\varepsilon (E| F)\). A similar result for spaces of germs of holomorphic functions on a compact subset is also established. Reviewer's remarks: (1) The \(\varepsilon\)-product was defined by Laurent Schwartz in a completely symmetric way. For Banach spaces, it was rediscovered later on by Waelbroeck. It is strange to read that the author quotes Waelbroeck's article for the fact that the \(\varepsilon\)-product is symmetric. (2) The correct title of reference [8] is `Quotient Banach spaces; multilinear theory' and the correct pages are 563--571. (3) Lines 4 and 5 of page 35 are repeated verbatim in lines 9 and 10 of the same page.