Recall the definition of weak convergence: if $u\_{k}\in \mathcal{B}$, we say that $u\_{k}$ converges weakly to $u$ if $\forall \ell \in $ the dual space $\mathcal{B}^{\*}$ ($\mathcal{B}$ is a Banach space) implies that $\langle \ell, u\_{k}\rangle \to \langle \ell, u\rangle$.

Now, for $u$ any fixed function in $C^{\infty}\_{0}(\mathbb{R})$ which is not identically 0, I need to show that the sequence of functions $u\_{k}(x)=u(x+k)$ converges weakly to $0$ in $L^{2}(\mathbb{R})$, but which does not converge in the usual sense.

The only problem is, I have absolutely no idea what this is supposed to look like - the only other problem I've ever seem like this was one where $u\_{k}$ was an orthonormal basis for the space it was in, so that was ridiculously easy to show. 

For that one, it basically involved something like showing that the sum of the inner product converged, and do that it's general term went to zero, but I'm not sure what that would look like in this case. If somebody would be willing to do this out in a detailed manner, just so I could see how a problem like this is supposed to be solved, I would appreciate it so very much!