Q:

Can we prove the weak convergence of sequence bounded by another weakly convergence sequence?

A series $a_n$ in some Hilbert space $H$ weakly converges to zero (meaning $f(a_n) \to 0$ $\forall f \in H^{'}$, the dual space of $H$). It is also given that for another sequence $b_n$, norm of each element is bounded like $\|b_n\| \leq C \|a_n\|$ for some $C>0$. Now can we conclude that $b_n$ converges weakly to zero as well ?

A:

No, take any sequence $\left\{a_n\right\}$ such that $a_n\to 0$ weakly and $\|a_n\|=1$ for all $n$. Then consider the sequence $\left\{b_n\right\}$ with $b_n=b$ for all $n$, where $b\neq 0$. Then 
$$\|b_n\|=\|b\|=\|b\|\|a_n\| $$
for all $n$, but clearly $\left\{b_n\right\}$ does not converge weakly to zero. 

