The gliding hump property in vector sequence spaces

Abstract

It is shown that vector sequence spaces with a gliding hump property have many of the properties of complete spaces. For example, it is shown that the//-dual of certain vector sequence spaces with a gliding hump property are sequentially complete with respect to the topology of pointwise convergence and also versions of the Banach-Steinhaus Theorem are established for such spaces. 9kn-l+l'' ''' kn "

Characterizations of a class of matrix transformations

“…If i,j E N with i :S j, let [i,j] Note that if O= X E TV S and E (X) is a vector subspace of s (X), then the 0-WGHP is just the WGHP was introduced in ([ll] or [15]). …”

On the Gliding Humps Property

“…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].…”

“….). 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, . .…”

“…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, for…”

“….). 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].…”