On the Gliding Humps Property

Abstract

We establish two uniform convergence results for duality pairs consisting of vector-sequence spaces with some gliding humps property and the corresponding function-sequence spaces. These uniform convergence results imply some important facts.Let X, Y be topological vector spaces and E(X) a vector space of X-valued sequences. For x ∈ E(X), let x k denote the k th coordinate of x and, hence, X) ] β be the family of function-sequences {f k } ⊆ Y X for which f k (0) = 0 for all k and the series X) ] β : each T k is linear and continuous} (see [10,11,13,14,15,17]). As usual, forThroughout this paper we assume that E(X) ⊇ c 00 (X) = {x = (x k ) ∈ X N : x k = 0 eventually} and E(X) is equipped by some Hausdorff vector topology which is stronger than the topology of coordinatewise convergence and, hence, E(X) is a K(X) space [3]. Let P n : E(X) → E(X) be the section map which sends (x 1 , x 2 , . . .) to (x 1 , . . . , x n , 0, 0, . . .). We say that E(X) has the property SUB if {P n } ∞ n=1 is uniformly bounded on bounded subsets of the domain space E(X) [13].Following D. Noll [10], we say that a sequence {z n } of nonzero vectors in E(X) is a block sequence if there is a strictly increasing {k n } ⊆ N such that z n = (0, . . . , 0, z n kn+1 , z n kn+2 , . . . , z n kn+1 , 0, 0, . . .). We say that E(X) has 0

Spaces of $\lambda$-Multiplier Convergent Series

“…The sequence space (λ, τ 0 ) is said to have the uniform convergent property if, for each σ(λ β , λ)-sequentially compact subset F of λ β and each B ∈ B, the series j u j t j converges uniformly with respect to (u j ) ∈ F and (t j ) ∈ B. [6] proved the following important conclusion: Theorem 7. Let c 00 ⊆ λ, (λ, τ 0 ) be a K-space and have the section uniform bounded property and the quasi 0-gliding hump property.…”