THE f-NORM; A GENERALIZATION OF THE NORM OF FUNCTIONAL ANALYSIS

Abstract

A generalisation,  f-norm, of the concept of norm as defined in Functional Analysis is introduced.  The norm of λx is |λ|times the norm of x .  In the definiton of the  f-norm, this requirement is relaxed, so that the f-norm of λx is  times the f-|λ|norm of , where f is a function satisfying certain properties, which are presented. Examples of f-norms are given, the metric induced by an f-norm is considered and results concerning continuity of certain functions obtained.  Sets that are convex with respect to an f-norm are studied.  Limits with respect to the  f-norm are also considered.