I am reading Rudin's Principle of Mathematical Analysis chapter 8, and did not learn anything about Lebesgue Integral. Throughout the questions, when I say Riemann integrable, I mean Riemann integrable on the interval $[-\pi, \pi]$.
Question 1: Is there any way for me to see that when $|f(x)|^2$ is Riemann integrable, then $f(x)e^{-inx}$ is Riemann integrable?
Question 2: Without using Lebesgue integral, is there any way for me to see that $e^{inx}/(2\pi)^{1/2}$ orthonormal system spans $L_2$ space?
In Rudin's book chapter 8, he talks about if a function is Riemann integrable, then its Fourier Series converges to the function in $L_2$ sense, but I am not sure how to show this when the function is just $L_2$ integrable and not Riemann integrable.
Thanks.
Best Answer
For question $(1)$, the claim is false. I think you really need to work with the Lebesgue integral for this to be true because if on $[-\pi,\pi],$ we take $D(x) = \begin{cases} -1& x \not\in \mathbb{Q}\\ 1 & x \in \mathbb{Q} \end{cases}$. Then, $D$ is not Riemann-integrable, but $|D|$ is.
On the other hand, $D$ is Lebesgue integrable and then the claim is true by Holder and the DCT.
For question $(2)$ what we just noted says that with the Riemann integral, $L_2$ is not an inner product space, at least, not via the integral, so you can't even define orthonormality.