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