Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 23H / a / Solution

Strong convergence means $\Vert f_j-f{\Vert }_p\to 0$; weak convergence means $\int f_{j}h\to \int fh$ for every $h\in L^q$. By duality choose $h$ of norm one with $\int fh$ arbitrarily close to $\Vert f{\Vert }_p$. Holder inequality and passage to the limit give $\Vert f{\Vert }_p\le \mathrm{liminf}\Vert f_j{\Vert }_p$.

Solved by gpt-5.6-sol high.