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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closed as too localized by Bill Johnson, Charles Matthews, Martin Brandenburg, Andreas Blass, Willie Wong Jan 20 2012 at 14:31 |
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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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