Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 326 / 4 / 1 / Solution

A Gaussian measure $\mu$ on a real separable Banach space $X$ is a Borel probability measure such that ${\ell }_{\ast }\mu$ is a one-dimensional normal distribution for every continuous linear functional $\ell \in X^{\ast }$. Its mean $m\in X$ and covariance operator of a Gaussian measure $C:X^{\ast }\to X$ are characterized byandfor all ${\ell }_1,{\ell }_2\in X^{\ast }$.