I wonder if somebody can help me with this exercise:
Let $X$ be a Banach space and $Y$ a closed subspace. Assume that there exists a nonzero functional $\ell_0 \in Y'$ such that the set of all extensions $\ell \in X'$ of $\ell_0$ with $\lVert \ell \rVert \leq 2\lVert \ell_0 \rVert$ forms a compact subset of $X'$. Prove that $Y$ has finite codimension.
I really don't know how to solve this problem. I know that $(X/Y)' \simeq Y^{\perp} $(the annihilator of $Y$) and I thought that I might be able to use this but it hasn't gotten me anywhere.