Every continuous function bounded implies compact

February 18, 2009

It occurs to me that it might be nice to post solutions to miscellaneous mathematical exercises at least once in a while.

I saw this one on a chalkboard earlier today; evidently the room was serving as the venue for an analysis class. It’s exactly the sort of elementary exercise that usually takes me a day to solve, if I’m lucky. But this time, I’m happy to report, I managed to figure it out in just a few minutes (while ostensibly listening to a lecture on something else).

Problem: Let U \subset \mathbf{R}^n be such that every real-valued continuous function on {}U is bounded. Prove that U is compact.

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The joy of “pathology”

April 12, 2008

Recently I had a conversation with a mathematician who had worked on the theory of Banach spaces early in his career, but had since left that particular subject. He explained that he had become disillusioned by the fact that “all the natural conjectures turned out to be false”; indeed, Banach spaces can have some “strange” properties, such as having uncomplemented subspaces, lacking a basis, admitting operators with no invariant subspaces, or admitting almost no operators at all. (Actually, in fact, they can even be unexpectedly well-behaved!) The last straw for this fellow, apparently, had been the Gowers-Maurey space (for which Gowers won the Fields Medal) , which has a whole bunch of “weird” properties.

Disparaging language is used with disturbing frequency by mathematicians to describe mathematical concepts. Examples are labeled “pathological”; objects are described as “badly behaved”; functions are called “nasty”; problems are said to be “ill-posed”. In a library once I encountered a book whose title actually was Differentiable Functions On Bad Domains — where “Bad” here is not the name of a mathematician, but the ordinary English word meaning the opposite of “good”.

You might think this is nothing but picturesque language — like calling a certain group “the Monster” — except that there are plenty of mathematicians who actually seem to think the way the labels suggest they do. The ex-Banach-spacer I mentioned above is only one example; spend some time among mathematicians and you will find many more. Indeed, such aversion to the unexpected has a distinguished historical pedigree, I am sorry to say. Who can forget the dismay with which Weierstrass‘s construction of a nondifferentiable continuous function was greeted? And even now there are still some people who are pissed off about Cantor’s discoveries, and who would sooner overthrow the standard axioms of mathematics than confront the “paradoxes” of the infinite. Even Hilbert, who had the good sense to regard infinitary mathematics as paradisal rather than paradoxical, nevertheless reacted with anger (!) to Gödel’s results on the limits of formalization, according to Constance Reid. One is reminded of the discoverer of the irrationality of \sqrt{2}, who, you will recall, was allegedly thrown overboard by his Pythagorean comrades.

I have never sympathized with this way of thinking. As far as I am concerned, unexpected “pathological” phenomena are a large part of what makes mathematics interesting in the first place. Indeed, this accords with the attitudes of other kinds of scientists with regard to their own fields. You generally don’t find physicists crying in agony about the discovery of black holes, or biologists resenting the existence of extremophiles. Why, then, do so many mathematicians insist on doing the equivalent?