Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 23H (Analysis of Functions)

For separable Banach $X$, the closed unit ball of $X^{\ast }$ is weak-star compact. Embed it in the product of compact discs by evaluations on a countable dense subset; diagonal subsequences converge on that subset, boundedness extends the limit to all $X$, and the limit functional remains in the ball. Metrizability of the bounded weak-star topology turns sequential compactness into compactness. Scaling gives the general Banach-Alaoglu theorem.

$H$ 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 $f_0$.