MAT425 Real Analysis L22

Prof. Thistleton

A Brief Introduction to Sequences and Series of Functions We know what it means to say that a sequence or series of numbers converges. In many applications we will deal with sequences and series of functions. For instance, you have already seen Taylor Series in Calculus, and may have seen Fourier Series in a course on Pde's.

There are two interesting questions which arise when considering a sequence of functions:

• If the functions which make up a sequence or a series have a certain characteristic (eg. they are differentiable, bounded, etc.), is it true that the limit function retains this character?
• May we operate on the elements of the sequence or series term by term (eg. differentiate, integrate) and pass this operation onto the limit?

We first define just what we mean by convergence in the case of functions.

Definition 101

A sequence of functions (f_n) defined on a set S is said to converge to the function f if, for all , .

Examples.

Uniform continuity is a useful way to characterize certain functions. We saw that when a function is continuous on a compact set then it is uniformly continuous there. Similarly, we will now define a type of convergence for sequences of functions, called uniform convergence, which will prove to be useful when we wish to preserve properties in the limit.

Definition 110

A sequence of functions (f_n) defined on a set S is said to converge to a limit f_n uniformly if, given , there exists a number N such that for all n > N and for all .

We will need to measure the distance between two functions. There are various ways to do this, many of which involve integrals. One simple way derives from the following.

Definition 113

Given a function define its sup norm, supremum norm, uniform norm as .

We are ready for a result.

Theorem 120

Let (f_n) be a sequence of continuous functions defined on S. If f_n converges to f uniformly, then f is continuous on S.