Sequence Spaces with Oscillating Properties

ISSN： 0022-247X

Source： Journal of Mathematical Analysis and Applications, Vol.200, Iss.3, 1996-06, pp. : 519-537

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Abstract

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⊂ W F while the other affords a sufficient condition for E⊂ F to imply E⊂ S F . Here F is any L varphi -space, a class of spaces which includes the class of separable FK-spaces, S F denotes the elements of F having sectional convergence, and W F 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 a sufficient condition for E beta , the beta-dual of E , to be sigma( E beta , E ) sequentially complete whereas the second assures both the weak sequential completeness of E beta and the AK-property for E with the Mackey topology of the dual pair ( E , E beta ).