Term for the space of annihilators

Question

I am learning about annihilators of vector spaces.

Let $V$ be a vector space, and $W\subset V$. The set of all linear functionals $a$ in the dual space $V^*$ such that $\forall w\in W,\,a(w)=0$ is denoted $W^0$ and $a$ is called an annihilator of $W$. If $W$ is a proper subset of $V$, then $\exists a\neq0$.

I have two questions:

1. I know that $a$ is termed an annihilator, but the space $W^0$ is termed... the annihilation space? What is/are the formal term(s) (I am happy to hear about equivalents, even archaic ones)

2. How should one properly denote $W^0$? I am writing W^0 but perhaps W^\circ is more appropriate - what is the correct way? Also, do any other notations exist?

Many thanks.