A sequence $(x_n)_n$ of a Banach Space $X$ is called basic if it is a basis of $\overline{span\{x_n\}}$. Prove that if $x_n\overset{w}{\to}0$ then $\|x_n\|\to 0$.
I was trying to define a $T\in X^*$ like if $y=\sum_{i=1}^{\infty}a_ix_i$ then $T(y)=\sum_{i=1}^{\infty}a_i\|y\|$ which would then imply the result, but there is no way to prove such a thing would be continuous. Any suggestions?