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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.

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  • $\begingroup$ The condition that $S$ is closed is to show that $\|\,\cdot\,\|_{B/S}$ is a norm and not just a seminorm. It is not used to show the completeness of the quotient space $\endgroup$
    – Federico
    Dec 7, 2018 at 19:56

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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 $\|f_n-f\|_{B/S}\leq \|f_n-f\|_B$.

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  • $\begingroup$ What does $l^1$ denote ? And since $B/S$ might be infinite dimensional , can we use the method of induction ? $\endgroup$
    – J.Guo
    Dec 7, 2018 at 19:41
  • $\begingroup$ $a\in\ell^1$ is just a way of saying that $\sum_n |a(n)|<\infty$. $\endgroup$
    – Federico
    Dec 7, 2018 at 19:45
  • $\begingroup$ "And since B/S might be infinite dimensional , can we use the method of induction ?" What do you mean? The induction is on $n$. Once you have decomposed $f_1,\dots,f_{n-1}$, you decompose $f_n$ in the required way $\endgroup$
    – Federico
    Dec 7, 2018 at 19:46
  • $\begingroup$ Sorry , I misunderstand the meaning of $l^1$ before. But I still don't see the point . By definition I can prove $\|f_n-f_{n-1}+h_n\|_B\leq\|f_n-f_{n-1}\|_{B/S}+a(n)$ ,but how to show that $\|f'_n-f_{n-1}'\|_B\leq\|f_n-f_{n-1}\|_{B/S}+a(n)$ $\endgroup$
    – J.Guo
    Dec 7, 2018 at 19:51
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    $\begingroup$ I wonder if this might be easier to understand decomposed into: (1) Apply the theorem that a normed v.s. $B$ is a Banach space iff whenever $\sum_{n=1}^\infty \lVert x_n \rVert < \infty$ then $\sum_{n=1}^\infty x_n$ converges in $B$. (2) If $\sum_{n=1}^\infty \lVert f_n \rVert_{B/S} < \infty$, choose $f_n' \in f_n + S$ s.t. $\lVert f_n' \rVert_B < \lVert f_n \rVert_{B/S} + a(n)$. (3) Then $\sum_{n=1}^\infty \lVert f_n' \rVert_B < \infty$ so $\sum_{n=1}^\infty f_n'$ converges in $B$, say to $f'$. (4) Show that $\sum_{n=1}^\infty (f_n+S)$ converges to $f'+S$ in $B/S$. $\endgroup$ Dec 7, 2018 at 20:01

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