What is the annihilator of null set?

Question

If $V$ be a vector space over a field of scalars $F$ and $S\subset V$. The Annihilator of $S$ is denoted by $S^o$ and defined by, $S^o=\{f\in V^*:f(\alpha)=0\text{ } \forall \alpha\in S\}$. $V^*$ is the dual space of $V$. If $S=\phi$ then $S^o$=?

Answer 1

If $S=\varnothing$, then $S^o=V^{*}$. Just apply the definition. Let $f$ be any member of $V^{*}$. Then $f(\alpha)=0\text{ }, \forall \alpha\in \varnothing$.

Also, from an intuitive perspective, as a set shrinks, its annihilator expands, and vice-versa.

Answer 2

One side is obvious, $S^{o}\subset V^*$. For the other case, Let $f\in V^*,f(x)=0,\forall x\in \phi$(Trivial)($\because \phi$ is an empty set).Hence, $V^*\subset S^{o}$.