compactness of a sequence in $W^{-1,q}$

Question

Let $\{T_n\}$ be sequence in $W^{-1,q}(\Omega)$ for $q\in(1,2]$ which satisfies $\langle T_n,\phi \rangle \rightarrow 0 \text{ for all } \phi \in W_0^{1,q}(\Omega).$

Can we conclude $\{T_n\}_{n\in \mathbb{N}}$ is pre-compact in $W^{-1,q}(\Omega)$? If so how to prove it?

P.S.: $\langle\cdot,\cdot \rangle$ denotes the duality bracket.

Answer 1

If it would be precompact, then your sequence would have a subsequence converging strongly towards zero. Your assumptions are equivalent to the weak convergence of $T_n$ towards zero. However, there are weakly convergent sequences which are not strongly convergent.