A projective description of weighted inductive limits

Abstract:

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 ${\mathcal {V}_0}C(X) = {\text {in}}{{\text {d}}_{n \to }}C{({V_n})_0}(X)$ can always be described by an associated system $\overline V = {\overline V _\mathcal {V}}$ of weights on $X$; the continuous seminorms on ${\mathcal {V}_0}C(X)$ are characterized as weighted supremum norms. If $\mathcal {V} = {\{ {\upsilon _n}\} _n}$ 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 ${\mathcal {V}_0}C(X)$, and which is also equivalent to ${\mathcal {V}_0}C(X)$ agreeing algebraically and topologically with the associated weighted space $C{\overline V _0}(X)$; 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.