Suppose $B$ is a Banach space and $S$ is a closed linear subspace of $B$ . The subspace $S$ defines an equivalence relation $f \sim g$ to mean $f-g \in S$ . If $B/S$ denotes the collection of these equivalence classes , then show that $B/S$ is a Banach space with norm $$\vert\vert f \vert\vert_{B/S} = \inf (\vert\vert f' \vert \vert_B \,\, , f'\sim f)$$
I can show that $B/S$ is a normed vector space. I want to proceed by show that for each $f_n$ , there exist a decomposition $f_n=f_n' + h_n$ with $h_n \in S $ and $f_n'$ form a cauchy sequence in $B$. If this has been proved , then let $f_n' \to f$ $$\vert\vert f-f_n \vert\vert_{B/S} \le \vert\vert f-f_n' \vert\vert_{B/S} + \vert\vert f_n'-f_n \vert\vert_{B/S} \le \vert\vert f-f_n' \vert\vert_B \to 0$$
But I have no idea how to do this , and I don't get the point how to use the condition $S$ is a 'closed' subspace.
Best Answer
Take $(f_n)_n$ which is Cauchy in $B/S$. By passing to a subsequence, we can assume that $\sum_{n} \|f_n-f_{n-1}\|_{B/S}<\infty$. Take a positive sequence $a\in\ell^1$. Inductively we can find decompositions $f_n=f_n'+h_n$ with $h_n\in S$ such that $\|f'_n-f_{n-1}'\|_B\leq\|f_n-f_{n-1}\|_{B/S} + a(n)$. But then $$ \sum_n \|f'_n - f_{n-1}'\|_B < \infty $$ which implies that $f_n' \to f$ for some $f\in B$.
Then you are done because $\|f_n-f\|_{B/S}\leq \|f_n'-f\|_B$.