quotients of Banach spaces by closed subspaces are Banach spaces under the quotient norm

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

Proof : In 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., .By definition of the quotient norm, there exists $x_n\in X_n$ such that

$$\parallel x_n\parallel \le {\parallel X_n\parallel }_{X/M}+2^{-n}$$

It is clear that and so, as $X$ is a Banach space, ${\sum }_n{x}_n$ isconvergent.

Let $x={\sum }_n{x}_n$ and $s_k={\sum }_{n=1}^k{x}_n$. We have that

$$x-s_k+M=(x+M)-(s_k+M)=(x+M)-\sum _{n=1}^k(x_n+M)=(x+M)-\sum _{n=1}^k{X}_n$$

Since ${\parallel x-s_k+M\parallel }_{X/M}\le \parallel x-s_k\parallel ⟶0$ we see that ${\sum }_n{X}_n$ converges in $X/M$to $x+M$. $\mathrm{\square }$