I am not able to do the following , could anyone help me?
$N$ be a closed vector subspace of a vector space $L$ such that $L/N$ finite dimensional, we need to show that any subspace of $L$ containing $N$ is closed.
Thank you.
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Can you find a relation between the subspaces of $L$ containing $N$ and the subspaces of $L/N$? – Daniel Fischer Aug 19 '13 at 14:24
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Let $K$ be a subspace of $L$ containing $N$. Then under $\pi : L \to L/N$ we see that $\pi(K)$ is a subspace of $L/N$ that is finite dimensional, and thus $\pi(K)$ being a subspace of a finite dimensional vector space is finite dimensional. Thus $\pi(K)$ is closed in $L/N$. Now the projection $\pi : L \to L/K$ is a continuous map and $\pi^{-1}(\pi(K)) = K$. Thus, $K$ is closed in $L$.