Let $H_1=(H_1,(\cdot, \cdot)_1)$ and $H_2=(H_2,(\cdot, \cdot)_2)$ be Hilbert space such that $H_1 \subset H_2$ and $H_1$ is continuously and densely embedded in $H_2$.
Question. If for some $u,v \in H_1$ holds $(u,v)_1=0$, then also $(u,v)_2=0?$
It's true under this hypothesis? Or under some additional hypothesis?
I think it's true, since $H_1$ is continuously and densely embedded in $H_2$. I was able to see this fact only for some particular cases. My interest is whether it is worth more generally, which I haven't been able to prove until then.
What is an embedding? Is it an isometric embedding? Because in that case this seems just true by definition. Is it just an injective linear map? Because in that case I see no reason why orthogonality should be preserved. Even just finite dimensional examples would work.