ABSTRACT
Sequences and series of functions play a pivotal role in modern mathematics, and also in engineering and physics. The theory of Fourier series, just as an example, was invented in the early nineteenth century as a means of decomposing a fairly arbitrary function into simple units. The more modern theory of wavelets takes the Fourier theory to new heights of beauty and power. Uniform convergence is a very powerful idea for guaranteeing that the limit of a sequence of functions is well behaved. The fact that continuity is defined with a limit, and that the limit of continuous functions need not be continuous, gives even more examples of situations in which limits do not commute. Of course a series of functions is understood by studying the sequence of its partial sums. So, in some sense, the theory of sequences of functions and the theory of series of functions are equivalent.
