First, a definition:
Let $X$ be a normed space. A subset $M (\neq \emptyset) \subset X$ is said to be total in $X$ if the span of $M$ is dense in $X$.
Now theorem 3.6-2 in Kreyszig states the following:
(a) If a subset $M \neq \emptyset$ of an inner product space $X$ is dense in $X$, then $M^\perp = \{ 0 \}$.
(b) If $X$ is a Hilbert space and if $M$ ( $\neq \emptyset$ ) $ \subset X$ such that $M^\perp = \{0\}$, then $M$ is total in $X$.
So far, so good!
Now my question is, what if $M^\perp = \{0\}$, but $X$ is not a complete inner product space? Can we find a non-trivial example of such a set $M$ which is not total in $X$?
