[Math] Closed set in normed vector space

Question

Is it true that a subspace M of a normed vector space X is closed if the limit of every sequence in M is contained in M? Whether or not X is complete? Are there alternative characterizations of closed?

Answer 1

"...the limit of every convergent sequence in $M$ is contained in $M$."

Yes, this is one definition of a set being closed in a metric space.

You seem to be concerned with the following case: Suppose $X$ is not complete, and $\{v_n\}$ is a Cauchy sequence in $M$ that does not converge. Does this automatically mean $M$ is not closed, because the limit of $\{v_n\}$ is not in $M$? Well no, because $\{v_n\}$ doesn't have a limit. In order to conclude that $M$ is not closed, you would need to exhibit a sequence $\{v_n\}$ in $M$ that does converge, but whose limit is not in $M$.

Another characterization of closed subspaces: $M$ is closed iff $M^C$ is open; i.e. for every $v \in M^C$ there exists $r>0$ so that if $||v-w||<r$, $w \in M^C$ as well.