The functional
$$
I:H^1(U)\to\mathbb R,
\qquad I(u)=\int_Uu
$$
is bounded by the <Cauchy-Schwarz inequality>. Hence $H=\ker I$ is a closed vector subspace of the Hilbert space $H^1(U)$. Every closed subspace of a Hilbert space is complete in the inherited norm, so $H$ is a Hilbert space with the standard $H^1$ inner product.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 105 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-105/2/a/solution