 Solution

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 $\int g_j{h}_j=1$.