Steve Reads

The Babe is studying for the math GREs, so I’m reviewing a lot of my undergraduate math education. She’s on linear independence in vector spaces, which is reminding me that I wasn’t really paying attention when we studied Fourier series in PDEs class. (Which is a shame: the professor, Jack Schaeffer, is the best math professor I had at CMU.) She’s using Serge Lang’s multivariable-calculus textbook, which does a great job integrating the linear algebra and generalizing for applications to Fourier series.

Now I’d like to dig more deeply into orthogonal functions. I have lots of questions:

• Are there any succinct results telling us, for instance, that the set of functions which have Taylor-series representations is a strict subset of the set of functions that have Fourier-series representations? It’s clear that the latter is is larger than the former, since the former can only represent very smooth functions, whereas the latter can even capture square waves.

• Are there series of orthogonal functions which allow us to represent even more functions than Fourier series do?

• Is the Taylor-series basis ({1, x, x², x³,  . . . }) orthogonal with respect to the standard inner product? (I.e., two functions are orthogonal if the integral of their product over some interval is zero.) Are they orthogonal with respect to some other interesting inner product?

• I’ve never studied infinite-dimensional spaces in general. How much of the linear algebra from finite-dimensional vector spaces carries over to the infinite-dimensional case? There seems to be a notion of a “dense span” that applies in the infinite-dimensional case, but not necessarily in the finite-dimensional one. The Schauder basis seems to be what I’m looking for. It’s not clear to me why one needs a totally different concept for a countable basis, when it seems like the natural extension of a finite basis is the limit of a sequence of finite bases. I’m surely missing something vital.

• Do uncountable bases ever come up?

• Can someone give me an example of a vector space that lacks a norm? A bit of googling didn’t help.

• So that’s cool: the completion of an inner-product space, with respect to the metric induced by the inner product, is a Hilbert space. I guess that’s a tautology, since a Hilbert space is just a complete normed vector space. Still, somehow I find that cool.

I think I have other questions, but I’ll leave it there for now. Probably the best summary here would be: can anyone point me to a book on the theory of this whole class of problems?