Brownian covariance from an integrated orthonormal basis ($k_i(t)=⟨1_{[0,t]},h_i⟩⟹K(s,t)=s\wedge t$)

For a complete orthonormal basis of $L^2({\mathbb{R}}_{+})$, the Parseval identity for a Hilbertian basis identifies the sum of the feature products with $⟨1_{[0,s]},1_{[0,t]}⟩=\min (s,t)$. The Cauchy-Schwarz inequality gives absolute convergence.