Some results on convergence closed function spaces

Abstract

P. Dienes (see, P. Dienes, The Taylor Series, Oxford, 1931) studied sequence and sequence space and contributed the notion of convergence closed sequence space. That is under what circumstances a sequence space will be called convergence closed. In fact he gave a set of conditions dealing with parametric limit and projective limit under the umbrella of which a sequence space will be convergence closed under a definition of convergence in sequence space. Later on he established a few results using the notion of convergence closed sequence space. Through these established results efforts has been made by him to exhibit the set of different conditions under which different sequence spaces can get the title of being convergence closed sequence space. Later on Sharan (see, L. K. Sharan, Some contributions to the theory of function spaces, Ph.D. Thesis, Magadh University, Bodh Gaya, Bihar, India, 1986) extended the notion of convergence closed for function space (or spaces). He investigated that a few function spaces suitably defined are convergence closed function spaces. In this paper our aim is to extend some results on convergence closed function space. In the course of the study of convergence closed function spaces the notions of parametric limit, projective limit, projective convergence and the dual space of a function space are used, but it is only for function spaces that we have also used the notion of the section of a function. Efforts are made by us here to establish a few results which show the fact that there are some function spaces which are convergence closed, while establishing these results the vital role played by the dual space of a function space is also discussed.