A projective description of weighted inductive limits. (English) Zbl 0599.46026

Considering countable locally convex inductive limits of weighted spaces of continuous functions, if \({\mathcal V}=\{V_ n\}_ n\) is a decreasing sequence of systems of weights on a locally compact Hausdorff space X, we prove that the topology of \({\mathcal V}_ 0C(X)=_{n\to}C(V_ n)_ 0(X)\) can always be described by an associated system \(\bar V=\bar V_{{\mathcal V}}\) of weights on X; the continuous seminorms on \({\mathcal V}_ 0C(X)\) are characterized as weighted supremum norms. If \({\mathcal V}=\{v_ n\}_ n\) is a sequence of continuous weights on X, a condition is derived in terms of \({\mathcal V}\) which is both necessary and sufficient for the completeness (respectively, regularity) of the (LB)-space \({\mathcal V}_ 0C(X)\), and which is also equivalent to \({\mathcal V}_ 0C(X)\) agreeing algebraically and topologically with the associated weighted space \(C\bar V_ 0(X)\); for sequence spaces, this condition is the same as requiring that the corresponding echelon space be quasinormable.
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 [Duke Math. J. 38, 379-385 (1971; Zbl 0214.378)], 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.


46E10 Topological linear spaces of continuous, differentiable or analytic functions
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
30H05 Spaces of bounded analytic functions of one complex variable


Zbl 0214.378
Full Text: DOI


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