Projective limits of weighted (LB)-spaces of continuous functions
The authors investigate the structure of countable projective limits of countably inductive limits, called (PLB)-spaces, of weighted Banach spaces of continuous functions. More precisely, given a weight \(v\) (strictly positive and continuous function) on a locally compact and \(\sigma\)-compact topological space \(X\), we define \[ Cv(X) = \{f\in C(X):\;||f||_v:=\sup_{x\in X}v(x)|f(x)| < \infty\} \] and denote by \(Cv_0(X)\) the closed subspace consisting on those functions \(f\) such that \(vf\) vanishes at infinity. Now we consider a double sequence \({\mathcal A}= \big((a_{N,n})_{n\in {\mathbb N}}\big)_{N\in {\mathbb N}}\) of weights such that, for all \(N,n\in {\mathbb N},\) \[ 0 < a_{N,n+1} \leq a_{N,n} \leq a_{N+1,n}. \] Then \[ {\mathcal A}_NC(X):= \text{ind}_{n\in {\mathbb N}}Ca_{N,n}(X),\;({\mathcal A}_N)_0C(X):= \text{ind}_{n\in {\mathbb N}}C(a_{N,n})_0(X) \] are \(LB\)-spaces. The object of study of this paper are the projective limits \[ {\mathcal A}C(X):= \text{proj}_{N\in {\mathbb N}}{\mathcal A}_NC(X),\;({\mathcal A}C)_0(X):= \text{proj}_{N\in {\mathbb N}}({\mathcal A}_N)_0C(X). \] The main results are Theorems 3.5 and 3.7, which characterize the vanishing of the derived projective limit functor for the weighted (PLB)-spaces \({\mathcal A}C(X)\) and \(({\mathcal A}C)_0(X)\) in terms of the defining double sequence of weights \({\mathcal A}.\) The barrelledness of the projective limits is discussed in Theorems 3.7 and 3.8. As shown in Section 4, this theory permits to treat not only the sequence spaces defined with sup norms but also the spaces of continuous linear operators from a Köthe echelon space into another or the tensor products of Fréchet and (LB)-spaces of null sequences.
scientific article; zbMATH DE number 5565275