If $X$ is a banach space and $M\subset X $
Define $M^{\perp} = \{x'\ :\ x'(x)=0\ \forall x\in M\}$ and $N^\perp = \{x\in X\ :\ x'(x)=0\ \forall x' \in N\}$ where $x'$ denotes the functional from the dual.
If $X$ is a normed space and $U$ is a closed suset of $X$, how do I show that there exist canonical isometric isomorphism, i.e. $(X/U)' \cong U^\perp $ and
$U'\cong X'/U^\perp$
I need help to learn this problem. Thanks.