# Tricky Orthogonal Complement Lemma?

I have the following definition: Let $$V$$ be an inner product space and let $$S \subseteq V$$. The orthogonal complement of $$S$$, denoted $$S^{\perp}$$, is the set $$S^{\perp} = \left\{\vec{v} \in V \mid \langle\vec{v}, \vec{x}\rangle = 0 \hspace{1mm} \forall\vec{x} \in S\right\}.$$ It seems that it may be possible that if $$S^{\perp} = \left\{\vec{0}\right\}$$ then $$\text{span} \hspace{0.2mm} S = V$$, but I'm having trouble proving if this is true or not. I've been trying to use the fact that $$S^{\perp} = \left(\text{span} \hspace{0.2mm} S \right)^{\perp}$$.

In general, we know that $$(S^{\perp})^{\perp} \subseteq S$$, where we always have equality in the finite-dimensional setting.
If $$S^{\perp} = \{0\}$$, then what is $$(\{0\})^{\perp}$$? What vectors satisfy $$\langle v, 0\rangle = 0$$?
Note that $$(S^{\perp})^{\perp} = (\{0\})^{\perp} \subseteq S\subseteq V$$, and so $$S=\dots$$