# annihilator subspace of normed space

Let $E_0$ be a subpace of the normed space $E$. Let $E_0^a = \{f \in E' : f(x)=0 \forall x \in E_0 \}$ $(E'=\mathcal{L}(E,\mathbb{K})$, where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C})$.

Show that $E_0^a$ is a closed subspace of $E'$.

Using the Hahn-Banach theorem, show that $E_0'$ and $E'/E_0^a$ are isometrically isomorphic.

For each $x\in E$ define $\kappa(x)\in E^{\prime \prime}$ by $\langle \kappa(x), f\rangle = \langle f, x\rangle$. Then $E_0^a = \bigcap_{x\in E} \ker \kappa(x)$ is closed as the intersection of a family of closed sets. (The kernels are closed by continuity of $\kappa(x)$.)