Covariance operator of a Gaussian measure

The covariance operator $K$ of a Gaussian measure on a real Hilbert space satisfies $⟨Kh,g⟩=\mathrm{Cov}(⟨U,h⟩,⟨U,g⟩)$. A positive self-adjoint operator is the covariance of a Hilbert-space-valued Gaussian random variable exactly when it is trace class.

If $K{e}_j={\lambda }_j{e}_j$ with ${\lambda }_j\ge 0$ and ${\sum }_j{\lambda }_j<\infty$, thenconverges in mean square and almost surely in the Hilbert space, and has Gaussian law $N(m,K)$.