Proof

By hypothesis, $[a,b]$ is a closed real interval, $p,q > 1$ are real numbers with $1/p+1/q=1,$ and $f,g:[a,b]\to\mathbb R$ are two Riemann-integrable functions.
Since $f,g$ are Riemann-integrable, so are the functions $fg,$ $f^p,$ and $g^q.$
Let $n\ge 1$ and $h:=\frac{b-a}n.$
The Riemann sum converges to the Riemann integral, for suitably big $n$ and the mash points $\xi_k:=h*k+a,$ $k=1,\ldots,n.$
According to the Hölder's inequality we get with the p-norms of the vectors $\phi:=(f(\xi_1),f(\xi_2),\ldots,f(\xi_n))$, $\psi:=(g(\xi_1),g(\xi_2),\ldots,g(\xi_n))$: $$\begin{eqnarray}\sum_{k=1}^n|f(\xi_k)g(\xi_k)|\cdot h&=&\sum_{k=1}^n|f(\xi_k)|h^{\frac 1p}\cdot |g(\xi_k)|h^{\frac 1q}\nonumber\\&\le& \left(\sum_{k=1}^n{|f(\xi_k)|^p}\right)^{\frac 1p}\cdot h^{\frac 1p}\cdot \left(\sum_{k=1}^n{|g(\xi_k)|^q}\right)^{\frac 1q}\cdot h^{\frac 1q}\nonumber\\&=&||\phi||_p||\psi||_q\cdot h.\nonumber\end{eqnarray}$$
For $n\to\infty$ we get with the integral p-norms the inequality $$\int_a^b |f(x)g(x)|dx\le ||f||_p||g||_q.$$
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Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition