Showing that $(Y, \| \cdot \|)$ is Banach, iff $Y \subset (X,\| \cdot \|)$ is closed.

Question

Exercise :

Let $Y$ be a subspace of the Banach space $(X, \| \cdot \|)$. Show that $(Y, \| \cdot \|)$ is Banach iff $Y$ is closed.

Question : Any tips or hints on how to start this ? I see myself to be lost, mostly due to me struggling with real analysis definitions, such as a topological space being closed. Even then, though, what would be a proper way to start or elaborate the proof asked ?

Answer 1

1. Suppose $Y$ is closed. Let $y_n$ be a Cauchy sequence in $Y$; show that it converges to a member of $Y$.
2. Suppose $Y$ is not closed. Find a Cauchy sequence in $Y$ that does not converge to a member of $Y$.

Answer 2

Simply use the fact that a subspace of a complete metric space is complete if and only if it is closed.