Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 23H / b

$H$ is a closed linear subspace of reflexive $L^p$. A minimizing sequence is bounded by the triangle inequality. Reflexivity supplies a weakly convergent subsequence with limit $f_0\in H$; each norm $\Vert f-g_i{\Vert }_p$ is weakly lower semicontinuous, so their sum attains the infimum at $f_0$.