I have a question about the proof that $X\backslash U$ is a Banachspace if $X$ is one and $U$ is closed.
In my book it is said, that for $x_k \in X$ and a series $\sum_{k=1}^{\infty}||[x_k]||< \infty$ and the idea is to show that this series without the norm has a limit. Therefore it is said that we can assume that we have $||x_k||\le ||[x_k]||+2^{-k}$ for every $k \in \mathbb{N}$ and I do not see why. Does anybody here know why this is true?
Since the quotient norm is defined as $\|[x]\| = \inf_{u \in U} \|x+u\|$, then for all $\epsilon >0$ , there is some $u \in U$ such that $\|x+u\| > \|[x]\| +\epsilon$. Since $u \in U$, we have $[x] = [x+u]$, and so $\|[x]\|= \|[x+u]\|$.
So given the sequence $x_k$ and $\sum_k \|[x_k]\|< \infty$, one can find $u_k \in U$ such that $\|x_k+u_k\| > \|[x_k]\| +2^{-k}$. If we let $x'_k = x_k+u_k$, then we have $x'_k$ with $\sum_k \|[x'_k]\|< \infty$.
The book is just saying that you might as well start with the $x'_k$ in the first place.