Let $X$ be an inner product space, and let $M$ be a non-empty subset of $X$. Then we have the following:
(a) If the space of $M$ is dense in $X$, then $M^\perp = \{0 \}$, that is, $x \in X$, $x \perp M$ imply $x =0$.
(b) If $X$ is complete and $M^\perp = \{0 \}$, then the span of $M$ is dense in $X$.
In (b), what if the space $X$ is not complete? In such a case, can we have a set $M$ such that $M^\perp = \{0\}$, but the span of $M$ is not dense in $X$?
