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
$$
\boxed{
Ch=\frac12\langle\varphi_1,h\rangle\varphi_1
+\frac12\langle\varphi_2,h\rangle\varphi_2}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-326/4/2/b/solution