Let $V$ be a complex vector space with two inner products $\langle\,\cdot\,,\,\cdot\, \rangle_1$ and $\langle\,\cdot\,,\,\cdot\, \rangle_2$ and suppose $$\langle v,v\rangle_1\leq\langle v,v\rangle_2$$ for all $v\in V$. In addition, let $H_i$ be the completion of $V$ with respect to $\langle\,\cdot\,,\,\cdot\, \rangle_i$.
Is it true that we can identify $H_2$ with a subset of $H_1$?
In the paper that I'm reading it is claimed that $H_2\subset H_1$ for the reason stated above in proposition 2.3 on page 33.