A projective description of weighted inductive limits

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.

Generalization of the B *‐algebra (C0(X),∥ ∥∞)

“…Most of them deal however with function spaces rather than algebras. See for example [3], [4], [9] or [10]. Correspondingly function algebras with the weighted topology have been considered in [1], [5], [12]- [15] or [18].…”

The subspace problem for weighted inductive limits of spaces of holomorphic functions.

“…In order to describe the topology of the weighted inductive limits VC(G) and VH(G), Bierstedt, Meise and Summers [BMS1] introduced the system of weights V , associated with the sequence V ,…”

“…Clearly the inclusions VC(G) ֒→ CV (G) and VH(G) ֒→ HV (G) are continuous. In [BMS1] it was proved that VC(G) = CV (G) and VH(G) = HV (G) hold algebraically and that the two spaces in each equality have the same bounded sets. Moreover one of the main results in [BMS1] shows that if V satisfies condition (S) (S) for all k there is l such that v l /v k vanishes at infinity on G then VH(G) = HV (G) holds topologically and VH(G) is a topological subspace of VC(G).…”

“…The aim of the present article is to solve in the negative a well-known open problem raised by Bierstedt, Meise and Summers in [BMS1] (see also [BM1]). We construct a countable inductive limit of weighted Banach spaces of holomorphic functions, which is not a topological subspace of the corresponding weighted inductive limit of spaces of continuous functions.…”

“…This corresponds exactly with the term "projective description" used by Bierstedt,Meise and Summers [BMS1] which is the one we will also utilize in this paper. In [BMS1] it was proved that countable weighted inductive limits of Banach spaces of holomorphic functions on arbitrary open subsets G of C N admit such a canonical projective description by weighted sup-seminorms whenever the linking maps between the generating Banach spaces are compact. This theorem extended previous work by B.…”