The w*-extension of a bounded linear functional

Question

Let Y be a Banach space and assume that $X$ is a $w^*$-closed subspace of $Y^*$. Let $f$ be a bounded linear functional on $X$. Does there exist any $w^*$-continuous linear functional $\phi$ on $Y^*$ with $\phi_{|_{X}}=f$?

Answer 1

No.

Consider the case $X=Y^*$. Your question is then whether any bounded linear functional on $Y^*$ must be weak* continuous, which is not so if $Y$ is not reflexive, since the weak* continuous linear functionals on $Y^*$ are precisely those corresponding to elements of $Y$.

Otoh if $Y$ is reflexive then yes: Any bounded linear functional on $X$ extends to a bounded linear functional on $Y^*$, and every bounded linear functional on $Y^*$ is weak* continuous, since $Y^{**}=Y$.