Dual pairs of sequence spaces. III

Abstract

In the previous papers [J. Boos, T. Leiger, Dual pairs of sequence spaces, Int. J. Math. Math. Sci. 28 (2001) 9-23; J. Boos, T. Leiger, Dual pairs of sequence spaces. II, Proc. Estonian Acad. Sci. Phys. Math. 51 (2002) 3-17], the authors defined and investigated dual pairs (E, E S ), where E is a sequence space, S is a BK-space on which a sum s is defined in the sense of Ruckle [W.H. Ruckle, Sequence Spaces, Pitman Advanced Publishing Program, Boston, 1981], and E S is the space of all factor sequences from E into S. In generalization of the SAK-property (weak sectional convergence) in the case of the dual pair (E, E β ), the SK-property was introduced and studied. In this note we consider factor sequence spaces E |S| , where |S| is the linear span of B S τ ω , the closure of the unit ball of S in the FK-space ω of all scalar sequences. An FK-space E such that E |S| includes the f -dual E f is said to have the SB-property. Our aim is to demonstrate, that in the duality (E, E S ), the SB-property plays the same role as the AB-property in the case E S = E β . In particular, we show for FK-spaces, in which the subspace of all finitely nonzero sequences is dense, that the SB-property implies the SK-property. Moreover, in the context of the SB-property, a generalization of the well-known factorization theorem due to Garling [D.J.H. Garling, On topological sequence spaces, Proc. Cambridge Philos. Soc. 63 (1967) 997-1019] is given.