My question is to determine the annihilator of the subset $$S=\{(x, y, z)\in \mathbb{R}^3 : xy = 1\}$$ of vector space $\mathbb{R}^3$.
Annihilator of a subset of a vector space can be stated as follows:
Annihilator of a subset $S$ of a vector space $V$ over a field $F$ is the subset $S^0$ of the dual space $V^*$given by $S^0 = \{f\in V^\star:f(\alpha)=0~~ \forall \alpha \in S\}$.
I fail to apply this definition to solve above stated problem. How to randomly search for such kind of functional. What should be a general approach to deal solve such kind of problem? I need help with this.
Thanks
If $\alpha\in S^0$, then $\alpha(1,1,1)=\alpha\left(2,\frac12,0\right)=\alpha\left(\frac12,2,0\right)=0$. Since $\left\{(1,1,1),\left(2,\frac12,0\right)\left(\frac12,2,0\right)\right\}$ is a basis of $\mathbb R^3$, $\alpha=0$. So, $S^0=\{0\}$.