On property in -spaces
In this paper we investigate conditions for a system of sequences of elements of a vector lattice; analogous conditions for systems of sequences of reals were studied by D. E. Peek.
Here we proved the existence of a closed vector space of sequences - any nonzero element of which does not comply with Schur’s property, that is, it is weakly convergent but not norm convergent. This allows us to find similar algebraic structures in some subsets of functions.
In normed linear space settings, modifying the sequential definition of continuity of an operator by replacing the usual limit "" with arbitrary linear regular summability methods we consider the notion of a generalized continuity (-continuity) and examine some of its consequences in respect of usual continuity and linearity of the operators between two normed linear spaces.