Non-natural topologies on spaces of holomorphic functions

Abstract

It is shown that every proper Frechet space with weak*-separable dual admits uncountably many inequivalent Frechet topologies. This applies, in particular, to spaces of holomorphic functions so solving in the negative a problem of Jarnicki and Pflug. For this case an example with a short self-contained access is added. It is a well known and often used fact, following from the closed graph theorem, that for any Frechet space of continuous functions the following is true: if any convergent sequence of functions in E also converges pointwise (or locally in L1) then convergence in E implies uniform convergence on compact sets. This is of particular interest if E is a Frechet space of holomorphic functions (see Krantz [2]). In this connection the question has been raised whether this might be true for any Frechet space E of holomorphic functions (see Jarnicki and Pflug [1, Remark 1.10.6, (b), p. 66]), that is, if every convergent sequence in E converges uniformly on compact sets. In the present note this question is solved in the negative in a very strict sense. For functional analytic tools and unexplained notation see [3]. The author thanks Peter Pflug for drawing his attention to this problem. Lemma 1 All proper (that is: not Banach) Frechet spaces with weak*-separable dual are linearly isomorphic. Proof: By dimE we denote the linear dimension of a linear space E and we set ω := CN with the product topology. By Eidelheit’s Theorem we know that there is a linear surjective map from E onto ω (see [3, 26.28]). This implies that dimω ≤ dimE. Let {y1, y2, . . . } be a weak*-dense set in E′. Then x 7→ (y1(x), y2(x), . . . ) is a linear, injective map E ↪→ ω. Therefore dimE ≤ dimω.