 Box norm

For nonempty finite sets $X,Y$ and a function $f:X\times Y\to \mathbb{C}$ on their Cartesian product, using uniform expectations, defineThis equals ${\mathbb{E}}_{y,y^{\prime }}|{\mathbb{E}}_{x}f(x,y)\overline{f(x,y^{\prime })}{|}^2$. Hence it is nonnegative, and vanishing forces $f=0$ by taking $y=y^{\prime }$. 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.