Fréchet and (LB) sequence spaces induced by dual Banach spaces of discrete Cesàro spaces (original) (raw)

Bulletin of the Belgian Mathematical Society - Simon Stevin

The Fréchet (resp., (LB)-) sequence spaces ces(p+) := r>p ces(r), 1 ≤ p < ∞ (resp. ces(p-) := 1<r<p ces(r), 1 < p ≤ ∞), are known to be very different to the classical sequence spaces ℓ p+ (resp., ℓ p-). Both of these classes of non-normable spaces ces(p+), ces(p-) are defined via the family of reflexive Banach sequence spaces ces(p), 1 < p < ∞. The dual Banach spaces d(q), 1 < q < ∞, of the discrete Cesàro spaces ces(p), 1 < p < ∞, were studied by G. Bennett, A. Jagers and others. Our aim is to investigate in detail the corresponding sequence spaces d(p+) and d(p-), which have not been considered before. Some of their properties have similarities with those of ces(p+), ces(p-) but, they also exhibit differences. For instance, ces(p+) is isomorphic to a power series Fréchet space of order 1 whereas d(p+) is isomorphic to such a space of infinite order. Every space ces(p+), ces(p-) admits an absolute basis but, none of the spaces d(p+), d(p-) have any absolute basis.