= Bilinear correlation bound for the box norm
For <real-valued functions> or complex <functions> on finite nonempty <sets>, with uniform <expectations>,
$$
|\mathbb E_{x,y}f(x,y)u(x)v(y)|\leq\|f\|_{\square}\|u\|_2\|v\|_2.
$$
To keep the complex case valid, retain every <complex conjugate>. Apply the <Cauchy-Schwarz inequality> first in $y$ and write
$$
|\mathbb E fuv|^2\leq\|v\|_2^2\,\mathbb E_{x,x'}u(x)\overline{u(x')}K(x,x'),\qquad K(x,x')=\mathbb E_y f(x,y)\overline{f(x',y)}.
$$
A second <Cauchy-Schwarz inequality> in $(x,x')$ bounds the last <expectation> by $\|u\|_2^2(\mathbb E_{x,x'}|K(x,x')|^2)^{1/2}$. The final factor is $\|f\|_{\square}^2$, by the <box norm> definition. Taking <square roots> proves the result. For <real-valued functions>, the <complex conjugates> can be omitted.
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