Closed Subspaces of a Banach Space are Banach

Last updated Nov 1, 2022

Let $(X, ||\cdot||)$ be a Banach Space. Suppose $C \subset X$ is a Closed Vector Subspace. Then $(C, ||\cdot|| {\big|}_{C})$ is a Banach Space.

Simple application of

1. $C$ is a Normed Vector Subspace
2. Closed Subset of a Complete Space is Complete

$\blacksquare$