A Projective Description of Weighted Inductive Limits

Abstract

Abstract. Considering countable locally convex inductive limits of weighted spaces of continuous functions, if "V = { V" )" is a decreasing sequence of systems of weights on a locally compact Hausdorff space X, we prove that the topology of %C( X) = ind"^ C(Vn)0(X) can always be described by an associated system V =■ V<v of weights on X; the continuous seminorms on °V0C( X) are characterized as weighted supremum norms. If CV= {v"}" is a sequence of continuous weights on X, a condition is derived in terms of "V which is both necessary and sufficient for the completeness (respectively, regularity) of the ( Lß)-space %C( X), and which is also equivalent to %C( X) agreeing algebraically and topologically with the associated weighted space CV0(X); for sequence spaces, this condition is the same as requiring that the corresponding echelon space be quasi-normable.A number of consequences follow. As our main application, in the case of weighted inductive limits of holomorphic functions, we obtain, using purely functional analytic methods, a considerable extension of a theorem due to B. A. Taylor [37] which is useful in connection with analytically uniform spaces and convolution equations.The projective description of weighted inductive limits also serves to improve upon existing tensor and slice product representations. Most of our work is in the context of spaces of scalar or Banach space valued functions, but, additionally, some results for spaces of functions with range in certain (LB)-spaces are mentioned.Introduction. Countable locally convex inductive limits of weighted spaces of continuous or, in particular, holomorphic functions naturally arise in great profusion throughout such fields as linear partial differential equations and convolution equations, distribution theory and representation of distributions as boundary values of holomorphic functions, complex analysis in one and several variables, and spectral theory and the holomorphic functional calculus. However, perhaps because the structure of general locally convex inductive limits is rather intricate, functional analytic applications have been restricted by a widespread belief that it was not possible to describe an inductive limit topology or the corresponding continuous seminorms in a setting which is general enough to include a substantial part of the applications, while at the same time, and above all, being "useful" for providing convergence results and direct estimates in terms of the seminorms. One aim of the

Generalization of the topological algebra (Cb(X), β)

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On locally convex extension of H∞in the unit ball and continuity of the Bergman projection

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Multiplication operators on weighted spaces in the non‐locally convex framework

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