I'm reading Induced Representations of locally compact groups by Kaniuth and Taylor and I don't understand how total subsets work (in particular in Lemma 2.24).
I know that a total subset of an inner product space is a set such that the closure of its linear span coincide with the entire space.
Shortly, in this lemma the authors want to prove that a particular subset $D$ of an Hilbert space $H$ is total. To prove this, they are showing that its linear span is dense in $V$, an inner product space which is dense in $H$ (actually $H$ was constructed as the completion of $V$).
Letting $v \in V$, it happens that $\langle d,v \rangle = 0$ for every $d\in D$ implies $v = 0$. By this last fact, the authors say that the span of $D$ is a linear dense subspace of $V$: this is what I can't understand. I looked for some informations on other books and it seems that in Hilbert spaces this can be viewed as an equivalent fact to being a total subset, but here we're considering $D$ a total subset of $V$ and not of $H$.
Is completeness necessary to prove this equivalence? I tried to prove it by myself, but I didn't manage to do it, nor I found books about this topic, since many of them (like Rudin's Real and complex analysis) introduce inner product spaces and switch to Hilbert spaces after a few pages. How does the fact that the orthogonal space of $D$ is trivial imply that $D$ is total?