Suppose \(X\) is a normed space . Then every complete subspace of \(X\) is closed.
Proof
Let \(X\) be a normed space, \(U\subseteq X\) a complete subspace and \(x_n\to x\) a convergent sequence in \(U\). Then \((x_n)\) is a Cauchy sequence
and therefore there is a limit in \(U\). Since limits are unique
, we have \(x\in U\) and \(U\) is therefore closed.