Riemann integrability question. This is what my professor put as "aside" and I have hard time understanding it:
Let $f,g$ be nonnegative bounded functions and $P,T$ be a partition pair.
$$\displaystyle \int_a^b f \bar{g} dx \le \left(\int_a^b |f|^2 dx \right)^{1/2} \left(\int_a^b |g|^2 dx \right)^{1/2}.$$
 A: I assume you know that if $f$ and $g$ are Riemann-integrable, then so is $f\cdot \overline{g}$. See here otherwise, p. 29. This justifies that we can take "limits" below to recover the integrals. So let $f,g$ be any Riemann-integrable functions over $[a,b]$.
Recall Cauchy-Schwarz for finite sequences. See here for the real case, from which the complex case follows easily by triangular inequality:
$$
\left|\sum_{j=1}^na_j\overline{b_j}\right|\leq\sqrt{\sum_{j=1}^n|a_j|^2}\sqrt{\sum_{j=1}^n|b_j|^2}.
$$
Now let $P=\{a=x_0< x_1<\ldots<x_n=b\}$ be a partition of $[a,b]$, with $x_j^*$ taken in each $[x_{j-1},x_j]$. We have
$$
\left|\sum_{j=1}^nf(x_j^*)\overline{g(x_j^*)}(x_j-x_{j-1})\right|=\left|\sum_{j=1}^nf(x_j^*)\sqrt{x_j-x_{j-1}}\overline{g(x_j^*)}\sqrt{x_j-x_{j-1}}\right|$$
$$
\leq\sqrt{\sum_{j=1}^n|f(x_j^*)|^2(x_j-x_{j-1})}\sqrt{\sum_{j=1}^n|g(x_j^*)|^2(x_j-x_{j-1})}.
$$
Passing to the "limit" as the partitions get finer, we end up with the Cauchy-Schwarz inequality for the Riemann integral, which is satisfied by every Riemann-integrable functions $f,g$ over $[a,b]$, and not only nonnegative ones:
$$
\left|\int_a^bf(x)\overline{g(x)}dx\right|
\leq \sqrt{\int_a^b|f(x)|^2dx}\sqrt{\int_a^b|g(x)|^2dx}.$$
A: This Is Just     Cauchy-Schwarz Inequality on $L^2$
