I try to describe my question: Let $H_1$ and $H_2$ be two completed space (may be Hilbert space). Here I say the topology on $H_1$ is stronger than topology on $H_2$ means that if a sequence $x_n$ converges strongly in $H_1$ then it also converges strongly in $H_2$. More precisely, $||u||_{H_1}>||u||_{H_2}$.
My question is: how can we derive from this property that a weakly convergence sequence in $H_1$ is also a weakly convergence sequence in $H_2$?.
The identity map from $H_1$ to $H_2$ is a bounded linear operator. It is a general fact that a bounded linear operator maps weakly convergent sequences to weakly convergent sequences.
Indeed, suppose $T:H_1\to H_2$ is bounded and $x_n\to x$ weakly. Let $T^*:H_2\to H_1$ be the adjoint of $T$. For any $y\in H_2$ we have $$ \langle Tx_n, y\rangle = \langle x_n, T^*y\rangle \to \langle x, T^*y\rangle = \langle Tx, y\rangle $$ proving that $Tx_n\to Tx$ weakly in $H_2$.