Rhapsody in Numbers

Another qual question, this one dealing with Hilbert spaces and the weak topology:

Exercise: Let \(\mathcal{H}\) be an infinite dimensional Hilbert space. Show that the unit sphere \(S := \{x\in\mathcal{H} : \|x\| = 1\}\) is weakly dense in the unit ball \(B := \{x\in\mathcal{H} : \|x\| \le 1\}\).

I find this to be a really surprising and counter-intuitive result. What this exercise asks us to prove is that every point inside of the unit ball in an infinite dimensional Hilbert space is, in some sense, arbitrarily close to the boundary of that ball. This is really striking to me—how can the center of a ball be really close to the boundary of that ball? Because this result is so unexpected, I think it is worth understanding. Indeed, the arguments presented below have helped me to build some better intuition about both infinite dimensional Hilbert spaces and the weak topology.

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In an undergraduate analysis class, one of the first results that is generally proved after the definition of differentiability is given is the fact that differentiable functions are continuous. We can justifiably ask if the converse holds—are there examples of functions that are continuous but not differentiable? If such examples exist, how “bad” can they be?

Many examples of continuous functions that are not differentiable spring to mind immediately: the absolute value function is not differentiable at zero; a sawtooth wave is not differentiable anywhere that it changes direction; the Cantor function¹I’ll (hopefully) talk more about this later. is an example of a continous function that is not differentiable on an uncountable set, though it does remain differentiable “almost everywhere.”

The goal is to show that there exist functions that are continuous, but that are nowhere differentiable. In fact, what we actually show is that the collection of such functions is, in some sense, quite large and that the set of functions that are differentiable—even if only at a single point—is quite small. First, we need a definition and an important result.

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The following problem is a fairly straightforward exercise that seems to come up fairly often on qualifying exams at UCR, and I think that I finally have a nicely argued proof. I am going to put it up here mostly for my own edification. Please note that the following is intended largely for an audience of advanced undergraduate or graduate students. I will be making no effort to give all of the required background, and assume some knowledge of the theory of Lebesgue integration and \(L^p\)-spaces.

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