Let $E$ be a linear space and $F\subset E$ a subspace.
If I equip the quotient space $E/F$ with a norm, it is required that $F$ is a closed subspace. Why is that required?
Normed Spaces – Understanding Norm on a Quotient Space
normed-spacesquotient-spaces
Best Answer
Let us first verify that this norm is well-defined in the case that $M \subset V$ is closed. If $\inf_{m \in M}\|x-m\|=0$, we know that there exists a sequence of $m_i \in M$ that converges to $x$. However, since $M$ was assumed to be closed, we know that $x \in M$, which is exactly the zero vector in our quotient space.
If it were true that such an $x \notin M$ ($M$ did not contain all of its limit points), there would be some $x \neq 0$ in the quotient with norm $0$, violating the nondegeneracy axiom.