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Let $X$ be an infinite dimensional vector space and $N$, $M$ be subspaces of $X$ such that $N \subseteq M$. Suppose $\dim(X/M)=\dim(X/N) \lt \infty$. Show that $$N=M$$ holds.

I know that above holds if $X$ is finite dimensional by using the dimension theorem to show $\dim(N)=\dim(M)$. However, I do not see a way in the infinite dimensional case.

Any help is appreciated.

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The canonical linear map $X/N \to X/M$, $x+N \mapsto x+M$ is obviously surjective and because of both spaces are of the same finite dimension it is also injective. This implies $M=N$.

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  • $\begingroup$ How does it imply $M=N$? $\endgroup$ – izimath Mar 26 '19 at 12:43
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    $\begingroup$ I think I got it. If $m \in M - N$, then $M \mapsto m+N$ which is impossible. Thanks! $\endgroup$ – izimath Mar 26 '19 at 12:47
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Hint: We have a natural surjective map $X/N\to X/M$.

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