# If $X$ is infinite dim'l and subspaces $N \subseteq M$ satisfy $\dim(X/M)=\dim(X/N) \lt \infty$, then $N=M$ holds.

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.

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