Annihilator of Subspace of Banach Space is Weak-* Closed
Theorem
Let $X$ be a Banach space.
Let $M$ be a vector subspace of $X$.
Let $X^\ast$ be the normed dual space of $X$.
Let $w^\ast$ be the weak-$\ast$ topology on $X^\ast$.
Let $M^\bot$ be the annihilator of $M$.
Then $M^\bot$ is closed in $\struct {X^\ast, w^\ast}$.
Proof 1
From Set is Closed iff Equals Topological Closure, we aim to show:
- $M^\bot = \map {\cl_{w^\ast} } {M^\bot}$
From Set is Subset of its Topological Closure, we have:
- $M^\bot \subseteq \map {\cl_{w^\ast} } {M^\bot}$
Now let:
- $f \in \map {\cl_{w^\ast} } {M^\bot}$
From Point in Set Closure iff Limit of Net, there exists a directed set $\struct {\Lambda, \preceq}$ and a net $\family {f_\lambda}_{\lambda \in \Lambda}$ in $M^\bot$ converging to $f$ in $\struct {X^\ast, w^\ast}$.
That is, from Characterization of Convergent Net in Weak-* Topology:
- for each $x \in X$ the net $\family {\map {f_\lambda} x}_{\lambda \in \Lambda}$ converges to $\map f x$ in $\GF$.
Since $f_\lambda \in M^\bot$ for each $\lambda \in \Lambda$, we have:
- $\map {f_\lambda} x = 0$ for each $x \in M$ and $\lambda \in \Lambda$.
From Constant Net is Convergent, we obtain:
- the net $\family {\map {f_\lambda} x}_{\lambda \in \Lambda}$ converges to $\map f x$ and $0$ in $\GF$.
From Metric Space is Hausdorff, $\GF$ is Hausdorff.
Hence from Characterization of Hausdorff Property in terms of Nets, we obtain $\map f x = 0$ for each $x \in M$.
So:
- $f \in M^\bot$
So we obtain:
- $\map {\cl_{w^\ast} } {M^\bot} \subseteq M^\bot$
and hence:
- $M^\bot = \map {\cl_{w^\ast} } {M^\bot}$
From Set is Closed iff Equals Topological Closure, we have that $M^\bot$ is closed in $\struct {X^\ast, w^\ast}$.
$\blacksquare$
Proof 2
From Annihilator of Subspace of Banach Space as Intersection of Kernels, we have:
- $\ds M^\bot = \bigcap_{x \in M} \map \ker {x^\wedge}$
From Characterization of Continuity of Linear Functional in Weak-* Topology:
- the linear functional $x^\wedge : \struct {X^\ast, w^\ast} \to \GF$ is continuous.
From Characterization of Continuous Linear Functionals on Topological Vector Space, $\map \ker {x^\wedge}$ is closed in $\struct {X^\ast, w^\ast}$.
So $M^\bot$ is the intersection of closed sets in $\struct {X^\ast, w^\ast}$, and hence is closed itself.
$\blacksquare$