I am making some elementary mistakes here. Could you please help me point out the problems? Thank you very much!
Suppose on some space $H$ we have two inner products, which make $H$ after completion two real Hilbert spaces. Suppose that these two inner products are comparable:
$$ (f,f)_1 \le (f,f)_2,\quad \forall f\in H. $$
Denote these two Hilbert spaces by $H_1$ and $H_2$. It is clear that
$$ H_2 \subseteq H_1. $$
Let $H_1'$ and $H_2'$ be the dual spaces of $H_1$ and $H_2$, respectively. They are also Hilbert spaces. Noticing that the fact that
$$ ||u||_{H_i'} = \sup_{(x,x)_i \le 1} |u(x)|,\qquad i=1,2, $$
we have that
$$ H_1' \subseteq H_2'\:. $$
Being Hilbert spaces, $H_i \cong H_i'$ (i.e., $H_i$ is isomorphic to $H_i'$). Since the Hilbert spaces are real, we can identify $H_i'$ by $H_i$. Then the above two inclusions imply that
$$ H_1 = H_2 , $$
which cannot be true in general. What is wrong in my arguments? Thank you very much for your great help! :-)
Anand