= 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 series>
$$
\boxed{
U=m+\sum_{n=1}^\infty\sqrt{\lambda_n}\,\xi_ne_n}.
$$
Because $\mathbb E\|U-m\|^2=\sum_n\lambda_n<\infty$, the series converges in $L^2(\Omega;X)$ and almost surely. Its law is the <Gaussian measure> $N(m,K)$.
Back to article page
