Let $Y_1$ and $Y_2$ be closed subspaces of a normed space $X$ (which may be infinite dimensional). How can I show that $Y_1$ and $Y_2$ must have different annihilators or else be the same subspace?
The annihilator $Y^0$ of a subspace $Y \subset X$ is defined as the set of functionals in $X'$ (the dual space of $X$) such that $f(y)=0\forall y \in Y, f \in Y^0$.
(Note: I have looked for similar questions, but all I have found are proofs that assume the space to be finite dimensional. I found a mention that this can be proved as a corollary to the Hahn-Banach theorem - how?)