Absolute boundedness and absolute convergence in sequence spaces

Abstract

Abstract.Let ft* be the set of all sequences h = (hk)k*Ax of Os and Is. A sequence x in a topological sequence space E has the property of absolute boundedness \AB\ if ft* • x = {y\yk = hkxk , h € ft*} is a bounded subset of E . The subspace E,AB, of all sequences with absolute boundedness in E has a natural topology stronger than that induced by E. A sequence x has the property of absolute sectional convergence \AK\ if, under this stronger topology, the net {h • x} converges to x , where h ranges over all sequences in ft* with a finite number of Is ordered coordinatewise (h1 < h" iff V/c, hk < hk ). Absolute boundedness and absolute convergence are investigated. It is shown that, for an F.K-space E, we have E = E,AB, if and only if E = l°° • E, and every element of E has the property \AK\ if and only if E = c0 • E . Solid hulls and largest solid subspaces of sequence spaces are also considered. The results are applied to standard sequence spaces, convergence fields of matrix methods, classical Banach spaces of Fourier series and to more recently introduced spaces of absolutely and strongly convergent Fourier series.

On l1-Invariant Sequence Spaces

“…As an indication of this fact we have the following observation that follows easily from the definitions; cf. [6].…”

“…Apart from the BK-spaces listed in [6], see also [27], the Köthe echelon spaces will turn out to be important for our work. Let A = a nk be a Köthe matrix, that is, a matrix of nonnegative numbers such that for all k ∈ there is some n ∈ with a nk > 0, and a nk ≤ a n+1 k for all n k ∈ .…”

“…A diagonal transform of a sequence space E is a space of the form E a = x ∈ ω x n a n ∈ E where a = a n is a sequence of nonzero scalars. As regards the notation for multipliers and sectional properties we shall follow the Goes-Buntinas school; see, for example, Section 2 of [6]. In particular, if E is a K-space containing ϕ then E AK is the set of all sequences x ∈ E that have property AK (that is, for which x = ∞ n=1 x n e n in E), E AB is the set of all sequences x ∈ ω that have property AB in E (that is, for which n k=1 x k e k n is bounded in E), and E AD denotes the closure of ϕ in E. The space E is said to have the property AK (AB, AD) if E = E AK E ⊂ E AB E = E AD , respectively .…”