Suppose $X$ is a Banach space with dual space $X^*$. If $Y$ is a closed subspace of $X$, then $Y^\perp=\{x^*\in X^*: x^*(y)=0 \text{ for all } y\in Y\}$ is a closed subspace in $X^*$. I am wondering whether this correspondence is reversible. That is, if $Y^*$ is closed in $X^*$, is it true that $Y^*\simeq (X/M)^*$ for some closed subspace $M$ in $X$?
Thank you.
Yes, it is. Let $M=\{x\in X: y^*(x)=0 \text{ for all }y^*\in Y^*\}$. This is sometimes denoted ${}^\perp Y^*$ and called the pre-annihilator of $Y^*$.
Every $y^*\in Y^*$ induces a linear functional on $X/M$ in a natural way: $y^*(x+M)=y^*(x)$ is well-defined.
Conversely, if $\phi$ is a linear functional on $X/M$, then its composition with the projection $\pi:X\to X/M$ defines $\phi\circ \pi$, an element of ${}^\perp Y^*$.
It is easy to see that two maps described in the two previous paragraphs are inverses of each other.