While writing the general case of the large sieve, one question of minor interest arose, which provides a nice exercise (or exam problem) for a course on finite-dimensional Hilbert spaces.

Since it’s not yet possible (as far as I can see, but I will try to investigate the issue) to include either LaTeX formulas (in the style used in a number of WordPress math blogs) or MathML formulas in the ETH blogs, I’ve resorted to the rather embarassing option of using `dvipng` to produce an image with the LaTeX content of this post…

Coming back to regular HTML, where one can make links, here’s one to the short note I wrote on this, with the proof of the result indicated. Note that I would be surprised is this were really at all new. There’s one lingering question that I haven’t answered at the moment: does there exist a proof by pure thought that, for the uniform density, there is an orthonormal basis of functions with constant modulus *1*?

## One thought on “An exercise with orthonormal basis”