A <Gaussian measure> $\mu$ on a real separable <Banach space> $X$ is a <Borel measure>[Borel probability measure] such that $\ell_*\mu$ is a one-dimensional <normal distribution> for every <continuous linear functional> $\ell\in X^*$. Its mean $m\in X$ and <covariance operator of a Gaussian measure> $C:X^*\to X$ are characterized by
$$
\ell(m)=\int_X\ell(u)\,d\mu(u),
$$
and
$$
\ell_2(C\ell_1)
=\int_X\ell_1(u-m)\ell_2(u-m)\,d\mu(u)
$$
for all $\ell_1,\ell_2\in X^*$.

Solution

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

For every $h\in L^2[0,1]$, the <continuous linear functional> induced by the <inner product> gives
$$
\langle U,h\rangle
=\xi_1\langle\varphi_1,h\rangle
+\xi_2\langle\varphi_2,h\rangle.
$$
This is a <normal distribution>[normal random variable] because it is a <linear combination of independent normal random variables>. Hence the law of $U$ is a <Gaussian measure>. Its mean is zero, and independence together with $\operatorname{Var}(\xi_i)=1/2$ gives
$$
\mathbb E[\langle U,h\rangle\langle U,g\rangle]
=\frac12\langle\varphi_1,h\rangle\langle\varphi_1,g\rangle
+\frac12\langle\varphi_2,h\rangle\langle\varphi_2,g\rangle.
$$
Thus its <covariance operator of a Gaussian measure> is
$$
\boxed{
Ch=\frac12\langle\varphi_1,h\rangle\varphi_1
+\frac12\langle\varphi_2,h\rangle\varphi_2}.
$$

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 326 / 4 / 2 / b / Solution

A continuous linear functional is a continuous linear map from a <topological vector space> to its scalar field. The collection of all such functionals is the <continuous dual space>.

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