How to prove that a closed subspace of a Banach space is a Banach space?
A subspace is closed if it contains all of its limit points. But in the proof of the above question how can use this idea to get a Cauchy sequence and show that it is convergent in the subspace?
Best Answer
Note: you don't need to "get a Cauchy sequence" -- you assume you have one and then you need to show that its limit is in the subspace.
Let $C$ be a closed subspace of a Banach space $B$, let $x_n$ be a Cauchy sequence in $C$ (with respect to the norm on $B$). Then the limit $x$ of $x_n$ exists in $B$ since $B$ is complete. Also, if it is not already in $C$, it must be a limit point of $C$. But $C$ is closed and hence contains all its limit points. Hence $C$ is complete, too.