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

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

Proof
We have: