# Hilbert space in Papa Rudin

In Rudin's Real and Complex Analysis, there is a problem in Chapter 4 on a Hilbert space $X = \text{span} \{e^{ist} \, \mid \, s \in \mathbb{R}\}$ with the inner product $$(f,g) = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} f(t) \overline{g(t)} \, dt.$$ It's completion evidently is a non-separable Hilbert space since $\{e^{ist}\}_{s \in \mathbb{R}}$ is an uncountable orthonormal basis.

My question is: is there a name for this space? Can anyone point me to references with more information about it? I've actually seen this inner product from time to time in signal processing (and I seem to remember it's related to the standard $L^{2}$ inner product, at least for some functions) and I suspect it would be useful to learn more about it.

• And what do you call his Functional Analysis, Grampa Rudin ? – Rene Schipperus Dec 31 '15 at 19:55
• I don't think that the functions which you are using for taking the span are orthonormal, at least not with the given scalar product. – PhoemueX Dec 31 '15 at 21:01
• $\frac{1}{2T} \int_{-T}^{T} e^{i(s - r)t} \, dt = \frac{1}{(s - r)T} \sin((s-r)T)$, which vanishes as $T \to \infty$ (assuming $s \neq r$). If $s = r$, it's $1$ for all $T$. – fourierwho Dec 31 '15 at 21:40
• You may have already noticed this, but Rudin mentions this exercise in "Notes and Comments" at the end of the book -- he calls the uniform limits of members of $X$ "almost periodic" – user363464 Dec 20 '17 at 6:18
• Thanks. I didn't realize the book had notes. – fourierwho Dec 20 '17 at 7:57