Theorem - Let $X$ be a Banach space and $M$ a closed subspace. Then $X/M$ with the quotient norm is a Banach space.

Proof : In order to prove that $X/M$ is a Banach space it is enough to prove that every series in $X/M$ that converges absolutely also converges in $X/M$ .

Let $\sum_{n} X_n$ be an absolutely convergent series in $X/M$ , i.e., $\sum_{n} \|X_n\|_{X/M} < \infty$ . By definition of the quotient norm, there exists $x_n \in X_n$ such that

It is clear that $\sum_{n} \|x_n\| < \infty$ and so, as $X$ is a Banach space, $\sum_{n} x_n$ is convergent.

Let $x = \sum_{n} x_n$ and $s_k = \sum_{n=1}^k x_n$ . We have that

Since $\|x-s_k + M \|_{X/M} \leq \|x-s_k\| \longrightarrow 0$ we see that $\sum_n X_n$ converges in $X/M$ to $x+M$ . $\square$