I went to a functional analysis course this year and my lecturer wrote down this Theorem. Lots of students pointed out it is incorrect, but she insisted it was. I am stating it now and hope someone can give me a good explanation about the confusion.
Here is the statement of the Theorem:
For any normed vector space $(V,\|\cdot\|_V)$, there exists another normed space $(Z,\|\cdot\|_Z)$ such that a map $i:V\rightarrow Z$ such that
(1) $i$ is an isometrical isomorphism
(2) $i(V)$ is dense in $Z$
where $Z$ is unique up to isomorphism.
In this Theorem, nothing said about $Z$ being complete. The students asked if $V$ is $\mathbb{Q}$, what $Z$ should be? $\mathbb{Q}$ fits the definition for $Z$, because $\mathbb{Q}$ is dense in $\mathbb{Q}$, but the lecturer said $\mathbb{Q}$ is not dense. She said something on the line dense means something different in a normed space. Can someone explain to me what this is about? In the Theorem, do we have to have say $Z$ is complete?