I am trying to understand why the operator $S$ below is unbounded.
We have $X \subset Y$ two Hilbert spaces, $X$ dense in $Y$ with continuous inclusion. Then we define $D(S)$ as being the set of $u$ such that the antilinear form $v \mapsto (u,v)_X$, $v \in X$ is continuous in the topology induced by $Y$.
Then we define $S$ by putting $(Su,v)_y = (u,v)_X$.
Question: why is this $S$ necessarily unbounded?
Can somebody give me any hint about how to justify it? I don't need a complete proof of the statement, I naturally want to do this by myself, I just need a hint...
Thank you very much people!