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 $X$.



weak*-closed subspaces
No. You get $Y^{\*\*}=Y^{\perp\perp}$ complemented in $X^{\*\*}$ and $Y$, being a dual space, is norm one complemented in $Y^{\*\*}$.


