I want to show that, given a subset $M$ of an Inner Product space $X$. If $M$ is a total set then, $M^\perp=\{0\}$. Which I have shown using the completion of $X$, which will be a Hilbert Space. And if a $A$ is dense in $X$ and $X$ is dense in $H$ then $A$ is dense in $H$.
I am trying another approach, where I clain $dim(V) = dim(\overline{V})$. (I've taken $V$ to be span of $M$). Is this claim always true? (I was posting my proof but I just found a flaw in it.)