Limits of Functions

Abstract

There are several reasonable definitions for the limit of a sequence of functions. Clearly the entries of the sequence should approximate the limit function f to greater and greater accuracy in some sense. But there are different ways of measuring the accuracy of an approximation, depending on the problem. Different approximation schemes generally correspond to different norms, although not all convergence criteria come from a norm. In this section, we consider two natural choices and see why the stronger notion is better for many purposes.