The FK-product E^F of two FK-spaces E and F is defined to be space of all sequences u which permit a representation « =x' • y' with x^ G E and y' S F, convergence of the series being coordinatewise, such that p{x']q{y') < oo for all continuous seminorms p on E and q on F. This definition is difFerent than that given by Buntinas and Goes but it is equivalent since E®F is also characterized as the smallest FKspace containing the coordinatewise product E-F = {x-y = (x^j/fc) \ x£ E, y £ f}. Some basic properties of FK-products are considered including characterizations of multiplier spaces. Examples of FK-products are given, including one which is a counterexample of a conjecture of Goes.1980 Mathematies Subject Classification. 46A45.
The concept of sectional convergence (AK) in FK-spaces was investigated by Zeller in (20). In (5) and (6), Garling investigated convergent and bounded sections in more general topological sequence spaces. Many of the results hold for Toeplitz sections in sequence spaces. A topological sequence space has the property of Toeplitz sectional convergence (TK) if and only if the unit sequences form a Toeplitz basis. In section 3, we present characterizations of Toeplitz sectional boundedness (TB) and functional Toeplitz sectional convergence (FTK) in terms of βT- and γT-duality. In section 4, we apply our results to summability fields. These results are related to the Hardy-Bohr property of multipliers for Cesàro summable sequences of positive order. In section 5, we characterize the properties TK and TB in FK-spaces by factorization statements.
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.
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