Lemma 6.1.12.label Let $X$ be a normed space and $Y\suf X$ is a closed subspace of finite codimension then
In particular if $X=Y\op Z$ then $X/Y\iso Z$ canonically and
Proof. Since $Y$ is closed the quotient $X/Y$ is a normed space with $\norm{x+Y}{}=\INF{x\in Y}\norm{x-y}{}$ so
\begin{align*}0=\norm{x+Y}{}=\INF{x\in Y}\norm{x-y}{}\iff x\in Y\iff x+Y=Y\end{align*}
$\dim(X/Y)=\infty$ hence $X/Y$ is complete by Lemma 6.1.10. We have the first assertion by Lemma 6.1.11. Since $X$ decomposes as a direct sum $X/Y\iso Z$ canonically hence we are done by considering $Z$ finite dimensional.$\square$