= Box Cauchy-Schwarz inequality
For four <real-valued functions> on a finite <Cartesian product>, let $\Lambda(f_{00},f_{01},f_{10},f_{11})=\mathbb E_{x_0,x_1,y_0,y_1}\prod_{i,j=0}^1 f_{ij}(x_i,y_j)$. Repeated <Cauchy-Schwarz inequalities> give
$$
|\Lambda(f_{00},f_{01},f_{10},f_{11})|\leq\prod_{i,j=0}^1\|f_{ij}\|_{\square}.
$$
First separate the two $y$ averages and apply <Cauchy-Schwarz inequality> in $(x_0,x_1)$. Each resulting squared factor is $\mathbb E_{y,y'}(\mathbb E_x f(x,y)f(x,y'))(\mathbb E_x g(x,y)g(x,y'))$, bounded by $\|f\|_{\square}^2\|g\|_{\square}^2$ by another <Cauchy-Schwarz inequality>. Expanding the four factors of $f+g$ and applying this inequality to each of the sixteen terms gives the <triangle inequality> for the <box norm>.
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