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Let $X$ be a Banach space.

Let $A$ and $B$ be closed subspaces of $X$.

Denote by $X/B$ the quotient space of $X$ by $B$. An element of $X/B$ is denoted by $\left[x\right]$. That is, $\left[x\right]=\left\{ x\right\} +B$.

Now, the question at hand:

Is $\left\{ \left[x\right]:x\in A\right\} $ a closed subspace of $X/B$?

(alternatively, one can consider $\text{dim}B<\infty$)

Joshua
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    Equivalently, you're asking whether or not the sum of two closed vector subspaces of a Banach space is closed. This is not true, see here for a counterexample in $L^2$. – Sassatelli Giulio Jan 09 '24 at 13:51
  • @SassatelliGiulio, can you explain the equivalence with the sum of closed subspaces? – Joshua Jan 09 '24 at 13:56
  • If you agree that, by whatever means you define the topology on $X/B$, you are considering the quotient topology, then it's just the definition: a subset of the quotient set is closed in the quotient topology if and only if its preimage by the quotient map is closed, and in your case said preimage is $A+B$. – Sassatelli Giulio Jan 09 '24 at 13:59
  • Thank you @SassatelliGiulio – Joshua Jan 09 '24 at 14:01
  • For the same reason, it is true if either of the subspaces is finite-dimensional (or if $B\cap A$ has finite codimension in either). – Sassatelli Giulio Jan 09 '24 at 14:18
  • I have found a proof of that assertion in https://math.stackexchange.com/questions/639570/sum-of-closed-subspaces-of-normed-linear-space – Joshua Jan 09 '24 at 14:58

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