 Box norm

For nonempty finite sets $X,Y$ and a real-valued function $f$ on their Cartesian product, using uniform expectations, defineThis equals ${\mathbb{E}}_{y,y^{\prime }}({\mathbb{E}}_{x}f(x,y)f(x,y^{\prime }){)}^2$. 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. This formula concerns real-valued functions; complex functions require appropriate complex conjugates in the definition.