Prove a density result in the usual Hilbert triple

Solution 1:

No, the solution is sadly incorrect.

One major mistake is in the step $$ (T-T_{w_k})(v) = (w-w_k,v)_V, $$ which is not true because you have defined $T_{w_k}(u) = (w_k,u)_H$ and not $T_{w_k}(u) = (w_k,u)_V$ (the latter would not allow you to show $T_{w_k}\in H^*$).

Note that there is no equality in any sense between $(\cdot,\cdot)_H$ $(\cdot,\cdot)_V$ (only continuity estimates for the induced norms).

Comments on your approach: I do not think your approach is likely to succeed. You start by picking a sequence $w_k\in V$ with $w_k\to w$. There, you might as well choose the constant sequence $w_k=w$. So, in this approach, there is nothing to be gained by considering the sequence $w_k$ (compared to considering just $w$).

A proof of the density is not so simple in my opinion, and some theory can be quite useful. An idea for a proof can be found here.