Annihilator of Subspace of Banach Space as Intersection of Kernels

Theorem

Let $\GF \in \set {\R, \C}$.

Let $X$ be a Banach space over $\GF$.

Let $M$ be a vector subspace of $X$.

Let $X^\ast$ be the normed dual space of $X$.

Let $M^\bot$ be the annihilator of $M$.

Then:

$\ds M^\bot = \bigcap_{x \mathop \in M} \map \ker {x^\wedge}$

where $x^\wedge$ denotes the evaluation linear transformation evaluated at $x$.

Proof

We have:

\(\ds M^\bot\)

\(=\)

\(\ds \set {g \in X^\ast : \map g x = 0 \text { for all } x \in M}\)

Definition of Annihilator of Subspace of Banach Space

\(\ds \)

\(=\)

\(\ds \bigcap_{x \mathop \in M} \set {g \in X^\ast : \map g x = 0}\)

\(\ds \)

\(=\)

\(\ds \bigcap_{x \mathop \in M} \set {g \in X^\ast : \map {x^\wedge} g = 0}\)

Definition of Evaluation Linear Transformation

\(\ds \)

\(=\)

\(\ds \bigcap_{x \mathop \in M} \map \ker {x^\wedge}\)

Definition of Kernel of Linear Transformation

$\blacksquare$