PARAMETRIC CONVERGENCE IMPLIES PROJECTIVE CONVERGENCE IN THE DUAL SPACE OF A FUNCTION SPACE

Abstract

Enormous work had been done on dual space of a sequence space. Also a massive corpus of work has been done in convergences of different kinds of sequence and sequence spaces. An account of all these can be found out in Cook,1 , chapter 10 P.272-326 . Later on dual space was extended to the case of function spaces. In this concern convergences has also been extended to the case of function spaces by some students of the school of mathematics. The role of dual space of a function space was also observed by some of the researcher of this field in the study of different discipline. In this paper we have used the notion of dual space of a function space to study the behavior of convergences in different suitably defined function spaces .That is we took the pain to establish some of the results on different kind of convergences for function spaces using the notion of a dual space of a function space. Moreover in course of extending some of the results we tried and got success in observing that if a result is true for a particular function space then it also stands good for another some of the function spaces. We also observed that the technique of establishing the results for function spaces are quite different from those of the technique of establishing the results to the case of sequence space .As a matter of fact the main reason behind it is the fact that in case of sequence space we had to deal with integers whereas in the case of function or function space, we always play with the continuous variables. The object of this paper is in fact to establish some of the results to show that parametric convergence implies projective convergence in the dual space of a function space.