Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 23H (Analysis of Functions)

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 $\Vert f{\Vert }_p\le \mathrm{liminf}\Vert f_j{\Vert }_p$.

Because $1<p<\infty$, $L^p$ is reflexive, so a bounded sequence has a weakly convergent subsequence $g_{j_k}\rightharpoonup g$. Thenand weak convergence handles $\int g_{j_k}h$. Thus the displayed limit holds (with the evident OCR correction $\int gh$ on the right).

Strong convergence of the tests is essential: in $L^2(0,2\pi )$ take the same orthonormal sequence for $g_j$ and $h_j$. Both converge weakly to zero, but $\int g_j{h}_j=1$.