The Problem With Pointwise Convergence

I wrote this up yesterday between my sections of college algebra, but forgot to post it afterwards. Oops.

We’ve got a problem with the topology of pointwise convergence. The subspace of continuous functions isn’t closed. What does that mean? It means that if we take a sequence of continuous functions, their pointwise limit may not be continuous.

Here’s an example in the real numbers. Let , which is a sequence of well-defined continuous functions on the entire real line. But if we take the pointwise limit we find that for , that for , and that for . So the functions in the sequence are continuous at , but the limiting function isn’t. It would be one thing if the sequence just failed to converge at some points — closedness doesn’t require all sequences to converge — but the pointwise limit clearly exists, and it fails to be continuous.

What we need is a stronger sense of convergence: one in which fewer sequences converge in the first place, and hopefully one in which the continuous functions turn out to be closed. But it should also obey the same definition as that of the pointwise limit when it does exist. And to find it we’ll need to recast the question of continuity in the limit.

Remember that a function is continuous at a point if it agrees with its limit there. That is, if . But the function should be the pointwise limit of the sequence : . And each of these functions is continuous: . Putting these together, the condition for continuity in the limit is

.

So our question is really about when we can exchange limits. For which sequences of functions do the dependence on and that on play well enough together to allow these limits to be exchanged? We’ll answer that question tomorrow.

Pointwise Convergence

When we evaluate a power series at a point we get a number if the series converges at that point. We even know that for each power series we have a disk where evaluation gives an absolutely convergent series at every point. In this view we regard a power series as the limit of a sequence of polynomials, evaluate each of the polynomials to get a sequence of numbers, and then take the limit of that sequence.

But what if we change it up. Let’s say we already know that our power series will converge with radius . Then inside the disk of radius each polynomial defines a function, and evaluation of the power series defines another function. It makes sense to regard the latter as “the limit” of the sequence of the former. That is, we already have as elements of the ring of power series . But now we regard them as living in the ring of complex-valued functions on the disk of radius .

And we have a topology on the ring of complex-valued functions on a domain . Instead of defining this topology in terms of open sets as we usually do, we define this topology in terms of which nets converge to which points. In fact, we’ll make do with sequences, since the extension to convergence of nets is straightforward.

The topology we have staring us in the face is the “pointwise” topology. That is, we say that a sequence of functions on converges to a function if and only if for every point the evaluations converge to the evaluation of : .

Alternately we can read this as a recipe: given a sequence of functions , if for each point the sequence of complex numbers converges, then we declare the limiting function to be that function defined by . If at any point the sequence fails to converge, we declare the sequence of functions to fail to converge.

Putting a topology on a space of functions marks the first dipping of our toes into the ocean of functional analysis. There’s a lot out there, and we’ll only be wading into the shallowest waters for now. Still, it gives a hint of the incredible depth that lays just beyond the breakers crashing on the shores of second-semester calculus.