Let $M$ be a closed subspace of a Banach space $X$. Let $q:X\to X/M$ be the quotient map, let $q^*: (X/M)^*\to X^*$ be the conjugate operator of $q$, and let $(z^*_n)$ be a sequence in $(X/M)^*$.
Is it true that $(z^*_n)$ is $w^*$-convergent in $(X/M)^*$ $\Leftrightarrow(q^* z^*_n)$ is $w^*$-convergent in $X^*$?
Are the following arguments correct?
The direct implication follows from the $w^*$-$w^*$ continuity of $q^*$.
For the converse, if $(q^* z^*_n)$ is $w^*$-null, then $z^*_n(q\, x) =(q^* z^*_n)(x)$ converges to $0$.