Sequence Spaces with Oscillating Properties

In this note we consider various types of oscillating properties for a sequence space E being motivated by an oscillating property introduced by Snyder and by recent papers dealing with theorems of Mazur]Orlicz type and gliding hump properties. Our main tools, two summability theorems, allow us to identify two such oscillating properties for a sequence space E one of which provides a sufficient condition for E;F to imply E;WF while the other affords a sufficient condition for E;F to imply E;SF . Here F is any Lw-space, a class of spaces which includes the class of separable FK-spaces, SF denotes the elements of F having sectional convergence, and WF denotes the elements of F having weak sectional convergence. This, in turn, is applied to yield improvements on some other theorems of Mazur]Orlicz type and to obtain a general consistency theorem. Furthermore, combining the above observations with the work of Bennett and Kalton we obtain the first oscillating property on a sequence space E as asufficient condition for Eb, the b-dual of E, to be s Eb, E. sequentially complete whereas the second assures both the weak sequential completeness of Eb and the AK-property for E with the Mackey topology of the dual pair E, Eb ..