In our lecture on functional analysis, we had the following statement: "Each normed space $X$ is isometrically isomorphic to a dense subset of a Banach space."
We were also given a proof, which goes as follows:
"Define $Y$ as the closure of $i(X) \subseteq X''$ (where $i: X \rightarrow X''$ is the linear isometry between a normed space $X$ and its bidual space $X''$). Since $Y$ is closed and $X''$ is a Banach space, we get that $Y$ is a Banach space."
I do understand the statements in the proof as such, but I don't understand how my statement actually follows from this proof. I think there might be just one final line of argument missing.
Thank you for your explanations.