The strict topology in certain weighted spaces of functions
This paper deals with abstract versions of theorems which compare various standard topologies on spaces of analytic functions. It continues previous work by the author [Sov. Math. Dokl. 25, 755-758 (1982; Zbl 0551.32003); translation from Dokl. Akad. Nauk SSSR 264, 827-830 (Russian) (1982)] on spaces of entire functions and, in a broader setting, work by \textit{K. D. Bierstedt}, \textit{R. Meise}, and \textit{W. H. Summers} [Trans. Am. Math. Soc. 272, 107-160 (1982; Zbl 0599.46026)], as well as others. The abstract setting consists of a non-empty set X (interpretation: open set in \({\mathbb{C}}^ n)\), an exhaustion of X by an increasing sequence of subsets \(X_ m\) (interpretation: \(X_ m\) compact, \(X_ m\subset int X_{m+1}\), \(\cup X_ m=X)\), the space \(F(X)\) of all complex-valued functions on X, and a linear subspace \(A(X)\) of \(F(X)\) (interpretation: space of analytic functions on X). Particularly important subspaces of \(F(X)\) are \(F_ 0(X)\), consisting of the bounded f such that for each \(\epsilon >0\), \(| f| <\epsilon\) off \(X_ m\) for some m (interpretation: functions that vanish at infinity), and \(F_ c(X)\), consisting of the bounded f that vanish off some \(X_ m\) (interpretation: functions with compact support). Topologies are introduced by means of systems of weights; we shall not define ''weight'' and ''system'' here. To a system of weights V is associated a linear subspace \(FV(X)\) of \(F(X)\), consisting of those f such that \(\| f\|_ v=\sup \{v\cdot | f| \}\) is finite for each \(v\in V\), and \(AV(X)=A(X)\cap FV(X)\); the seminorms \(\| \cdot \|_ v\) make these into locally convex spaces, and certain natural continuity conditions hold. If \({\mathcal V}=\{V_ m\}\) is a sequence of systems of weights, both a new system of weights \(\bar V\) and a new sequence of systems of weights \({\mathcal W}=\{W_ m\}\) associated to \({\mathcal V}\) are defined, and under appropriate naturality conditions the inductive limits \({\mathcal V}A(X)=\lim ind\;AV_ m(X)\) and \({\mathcal W}A(X)=\lim ind AW_ m(X)\) can be defined. There is a natural continuous embedding of \({\mathcal W}F(X)\) into \(F\bar W(X)\), and the first theorem is that this embedding is a topological isomorphism (into, in general). The space \({\mathcal V}A(X)\) has three topologies: its own topology \(\tau\), a topology induced from the topology \(\mu\) of \(F\bar V(X)\), and a topology \(\beta_ 1\) induced from the topology of \({\mathcal W}F(X)\). \(\tau\) is the strongest and \(\beta_ 1\) the weakest of these. The rest of the paper concerns developing hypotheses that force these topologies to coincide. Theorem 4 provides the desired conclusion if (\({\mathcal V}A(X),\tau)\) is semi-Montel (every bounded set is relatively compact) and each \(FW_ m(X)\) is of type gDF (that is, has a sequence \(\{B_ n\}\) of bounded sets such that each bounded set is absorbed by some \(B_ n\), and the given topology is the strongest locally convex topology which coincides with the given topology on each \(B_ n)\). Comparisons to some other topologies are given.
scientific article; zbMATH DE number 3996390
scientific article