Let $T$ be an unbounded self-adjoint operator.

Does there exist, for any $\varphi$ normalized in the Hilbert space, a constant $k(\varphi)>0$ and a sequence of normalized $(\varphi_n)$ such that $$ \lim_{n \rightarrow \infty} \Vert \varphi-\varphi_n \Vert=0 $$ and $\Vert T \varphi_n \Vert \le k(\varphi).$

Somehow this looks strange, if you think of $\varphi \notin D(T)$ as then $$\Vert T\varphi \Vert"="\infty$$

all of a sudden, on the other hand, maybe you only have to choose $k$ in a suitable way.