= Box norm
{title2=$\|f\|_{\square}$}
For nonempty <finite sets> $X,Y$ and a <real-valued function> $f$ on their <Cartesian product>, using uniform <expectations>, define
$$
\|f\|_{\square}^4=\mathbb E_{x,x',y,y'}f(x,y)f(x,y')f(x',y)f(x',y').
$$
This equals $\mathbb E_{y,y'}(\mathbb E_xf(x,y)f(x,y'))^2$. 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>. This formula concerns <real-valued functions>; complex <functions> require appropriate complex conjugates in the definition.
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