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

Let $X$ be a Banach space and $M$ a closed subspace. Then $X/M$ with the quotient norm is a Banach 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., $\sum_{n}\|X_{n}\|_{X/M}<\infty$. By definition of the quotient norm, there exists $x_{n}\in X_{n}$ such that

 $\|x_{n}\|\leq\|X_{n}\|_{X/M}+2^{-n}$

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

 $x-s_{k}+M=(x+M)-(s_{k}+M)=(x+M)-\sum_{n=1}^{k}(x_{n}+M)=(x+M)-\sum_{n=1}^{k}X_% {n}$

Since $\|x-s_{k}+M\|_{X/M}\leq\|x-s_{k}\|\longrightarrow 0$ we see that $\sum_{n}X_{n}$ converges in $X/M$ to $x+M$. $\square$

Title quotients of Banach spaces by closed subspaces are Banach spaces under the quotient norm QuotientsOfBanachSpacesByClosedSubspacesAreBanachSpacesUnderTheQuotientNorm 2013-03-22 17:23:01 2013-03-22 17:23:01 asteroid (17536) asteroid (17536) 7 asteroid (17536) Theorem msc 46B99