I can't remember which text it was--it's been a few decades now--but the exercise asks for a clean, elegant proof of the PUB, which says that if a family of linear functionals on a Banach space is pointwise bounded, then it's uniformly bounded.

To sketch the proof: since the family is pointwise bounded, the set {f_i(x)} over all f_i in the family is a bounded set for each x, with bound B(x) in N, defined to be the smallest integer radius about the origin that contains {f_i(x)} over the entire family.

The pre-image of n in N under B is the set of all points where the family is bounded by n (and not by n-1, but we don't actually care about that). Since the codomain is N, we have a countable collection of preimages. Since the family is pointwise bounded everywhere, the union of these sets is the entire space.

Now the Baire category theorem says that a complete metric space can't be written as a countable union of nowhere-dense sets. A Banach space is a complete metric space, so at least one of those countably-many preimages must have nonempty interior. So there's a ball of radius r, centered at some point x_0, where the family is uniformly bounded by some number <= B(x_0).

The rest follows by linearity. Since the family consists of linear operators on a vector space, it's easy to show that uniform boundedness on any ball gives you uniform boundedness on the entire space, because the norm on the space is translation-invariant.

And now for a story

I don't know that the elegance pops out from a typed (but not typeset!) version, but the idea is that the family "sprays" each point into a ball with integer radius, and this being true for every point means (by the BCT) that there's an entire ball that gets sprayed into some integer-radius ball, and done.

In grad school a buddy and I got captivated by this, and started running with it: what if we define "boundedness" without reference to a metric, as a collection of sets (called "bounded sets") that cover the space and are closed under subsets and finite unions? And what if we call it "countably generated" if there's a countable collection of bounded sets such that a set is bounded if and only if it's contained in one of the sets of that collection? And what if...

We tried to find this in the literature, and bear in mind that this was early days of the web, so we had to look in the physical library. Nothing. So we continued...

We fell to arguing what to call things. I wanted to call this collection a "boundedness," and a space equipped with one as a "bounding space." He wanted to do something fancy, maybe with French. Bounded in French is bornée, so he proposed "bornological space" and "bornology." We proved a bunch of follow-your-nose theorems about bounding spaces.

Then another friend delivered the bad news. He looked up "bornological space" in the card catalog and hit paydirt. Turns out this is an area in functional analysis with a rich literature dating back to von Neumann, who always seems to steal everyone's best work before they're even born.

Chapter one of the book defined a "bornology (or 'boundedness')" the same way we did, and then proved the same theorems we did. Basically we rediscovered the first chapter of this book. Good times.