Property (BB) and holomorphic functions on Fréchet-Montel spaces

Abstract

Two of the most important topologies on the space ℋ( E ) of all holomorphic functions f : E → ℂ on a complex locally convex space E are the compact-open topology τ 0 and the Nachbin-ported topology τ ε . We recall that a seminorm p on ℋ( E ) is said to be τ ω -continuous if there is a compact K such that for every open set V with K contained in V there is a constant c > 0 satisfying Clearly, the following natural question arises: when do the topologies τ 0 and τ ω coincide? In the setting of Fréchet spaces equality of τ 0 and τ ω forces E to be a Montel space; Mujica [ 21 ] proved τ 0 = τ ω for Fréchet-Schwartz spaces and Ansemil-Ponte [ 1 ] showed that for Fréchet-Montel spaces this happens if and only if the space of all continuous n -homogeneous polynomials with the compact-open topology, (P n ( E ), τ 0 ), is barrelled. By duality, it turns out that the question τ 0 = τ ω is intimately related to ‘Grothendieck's problème des topologies’ which asks whether or not for two Fréchet spaces E 1 and E 2 each bounded set B of the (completed) projective tensor product is contained in the closed absolutely convex hull of the set B 1 ⊗ B 2 , where B k is a bounded subset of E k for k = 1, 2. If this is the case, then the pair ( E 1 , E 2 ) is said to have property (BB). Observe that every compact set B in can always be lifted by compact subsets B k of E k (see e.g. [ 20 ], 15·6·3). Hence, for Fréchet-Montel spaces E 1 and E 2 , property (BB) of ( E l , E 2 ) means that is Fréchet-Montel and vice versa. Taskinen[ 24 ] found the first counterexample to Grothendieck's problem. In [ 25 ] he constructed a Fréchet-Montel space E 0 for which ( E 0 , E 0 ) does not have property (BB), and Ansemil-Taskinen [ 2 ] showed that τ 0 ≠ τ ω on ℋ( E 0 ).