= Box norm
{title2=$\|f\|_{\square}$}
= Four-cycle norm
{synonym}
For nonempty <finite sets> $X,Y$ and a <function> $f:X\times Y\to\mathbb C$ on their <Cartesian product>, using uniform <expectations>, define
$$
\|f\|_{\square}^4=\mathbb E_{x,x',y,y'}f(x,y)\overline{f(x',y)}\overline{f(x,y')}f(x',y').
$$
This equals $\mathbb E_{y,y'}|\mathbb E_xf(x,y)\overline{f(x,y')}|^2$. Hence it is nonnegative, and vanishing forces $f=0$ by taking $y=y'$. Absolute homogeneity and the <box Cauchy-Schwarz inequality> establish that it is a <norm>. For a <real-valued function> the complex conjugates can be omitted. The <bilinear correlation bound for the box norm> also holds for complex functions.
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