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## Is the quotient of closed subspaces in a Banach space again closed? [closed]

Let $X$ be a Banach space, $V$ and $W$ two closed subspaces of $X$ with $W \subset V$. Is it true that $V/W$ is closed in $X/W$?

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What are the continous linear functionals that vanish on it? – Charles Matthews Jan 20 2012 at 13:49
This is a trivial test if one has understood the quotient topology. – Martin Brandenburg Jan 20 2012 at 13:51

## 1 Answer

Yes, it is closed. Proof: The quotient map $p: X\to X/W$ is open. $U=X\setminus V$ is open. So is $p(U)=(X/W)\setminus (V/W)$.

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