TITLE: weak*-closed subspaces
QUESTION [7 upvotes]: Recall that a closed subspace $Y$ of a Banach space $X$ is weakly complemented if the set 
$$Y^{\bot}:= \{ f\in X^\*| f(y) = 0 \forall y\in Y\}$$ 
is a complemented subspace of $ X^*$. For example, $c_0$ is a weakly complemented subspace of $l_{\infty}$.
Question: Is there a Banach space $X$ such that there is a weak${}^*$-closed subspace $Y$ which is weakly complemented but not complemented in $X$.

REPLY [7 votes]: No. You get $Y^{**}=Y^{\perp\perp}$ complemented in $X^{**}$ and $Y$, being a dual space, is norm one complemented in $Y^{**}$.