 Solution

The compact self-adjoint operator $K$ has an orthonormal basis of eigenvectors $e_n$. Part (c) gives ${\lambda }_n>0$ and ${\sum }_n{\lambda }_n<\infty$, so $K$ is positive and trace class, exactly the required condition for a covariance operator of a Gaussian measure on a Hilbert space.

For independent standard normal variables ${\xi }_n$, define the Hilbert-space Gaussian seriesBecause $\mathbb{E}\Vert U-m{\Vert }^2={\sum }_n{\lambda }_n<\infty$, the series converges in $L^2(\Omega ;X)$ and almost surely. Its law is the Gaussian measure $N(m,K)$.