WEIGHTED (LF)-SPACES OF CONTINUOUS-FUNCTIONS

In the present article weighted inductive limits [GRAPHICS] of Frechet spaces of continuous functions on a locally compact (Hausdorff) space X are studied, where V(n) = (v(n,k))k is-an-element-of N and the weights v(n,k) are positive and continuous on X, v(n,k) less-than-or-equal-to v(n,k+1) and v(n,k) greater-than-or-equal-to v(n+1,k) for n, k = 1,2, ... . We investigate the behaviour of the bounded subsets and the completeness of VC(X) and V0C(X), introduce the ''projective hulls'' CVBAR(X) and CV0BAR(X) and characterize when VC(X) = CVBAR(X) resp. V0C(X) = CV0BAR(X) holds, thus providing a description of the continuous seminorms in the weighted inductive limits (by means of weighted sup-seminorms).