Analysing Smooth Bump Functions and Convex Variational Principles | Original Article

ABSTRACT:

In the existing paper we check out the attributes of separable Banach areas admitting bump functions. As for uses of smooth bump functions, in this paper we mainly concern ourselves with smooth partitions as well as approximation of unity as it ties in effectively with the main design of ours. We shall think about a variety of ailments on a Banach area X which ensure the presence of smooth partitions of unity. While the presence of a smooth bump function on X is clearly an essential state for the presence of smooth partitions of unity on X, the question of sufficiency is a great deal of standing wide open issue. To this conclusion, we create some fundamental concept of functions on separable Banach spaces.