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

The functionalis 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.