Cauchy-Schwarz for Multiple Integrals Is there a generalization of the Cauchy-Schwarz Inequality for multiple integrals?
 A: According to Michael Steele, one generalization is the following for double integrals: $S \subset \mathbb{R}^2, f: S \to \mathbb{R}$ and $g: S \to \mathbb{R}$, then $$A = \iint_{S} f^{2} \ dx \ dy, \ B = \iint_{S} fg \ dx \ dy, \ C = \iint_{S} g^{2} \ dx \ dy$$ satisfy $|B| \leq \sqrt{A} \cdot \sqrt{C}$.
A: There is an exact generalisation: for any space of functions over a measure space $(X,\Sigma,\mu)$ where $ \int_X \lvert u \rvert^2 \, d\mu = 0 $ if and only if $u =0$, then if we write $\langle u,v \rangle = \int_X \bar{u}v \, d\mu$, we have
$$ \langle u,v \rangle^2 \leqslant \langle u,u \rangle \langle v,v \rangle, $$
with equality if and only if $u=kv$ for some constant $k$.
If $\langle u,u \rangle=0$ or $\langle v,v \rangle=0$, this is clear. Suppose now neither is zero. Then
$$ \frac{\lvert \langle u,v \rangle \rvert}{\sqrt{\langle u,u \rangle \langle v,v \rangle}} = \left\lvert \int_X \frac{\bar{u}}{\sqrt{\langle u,u \rangle}} \frac{v}{\sqrt{\langle v,v \rangle}} \right\rvert \, d\mu \leqslant \left\lvert \frac{1}{2}\int_X \left( \frac{\lvert u \rvert^2}{\langle u,u \rangle} +  \frac{\lvert v \rvert^2}{\langle v,v \rangle} \right) \right\rvert \, d\mu = 1, $$
using the elementary AM-GM inequality $ 2\lvert ab \rvert \leqslant \lvert a \rvert^2+ \lvert b \rvert^2$. It is clear that equality holds precisely when the functions are proportional, by the conditions of the AM-GM inequality.

A more direct way in this case is to note
$$ 0 \leqslant \frac{1}{2}\iint \lvert u(x)v(y)-u(y)v(x) \rvert^2 \, d\mu(x) \, d\mu(y) = \langle u,u \rangle \langle v,v \rangle-\lvert \langle u,u \rangle\rvert^2, $$
obviously with equality only if $u(x)v(y)-u(y)v(x) = 0$, so $u(x)=kv(x)$.
