 Bilinear correlation bound for the box norm

For real-valued functions or complex functions on finite nonempty sets, with uniform expectations,To keep the complex case valid, retain every complex conjugate. Apply the Cauchy-Schwarz inequality first in $y$ and writeA second Cauchy-Schwarz inequality in $(x,x^{\prime })$ bounds the last expectation by $\Vert u{\Vert }_2^2({\mathbb{E}}_{x,x^{\prime }}|K(x,x^{\prime }){|}^2{)}^{1/2}$. The final factor is $\Vert f{\Vert }_{\square }^2$, by the box norm definition. Taking square roots proves the result. For real-valued functions, the complex conjugates can be omitted.