Closed Subspaces of Vector Spaces

Question: In Functional Analysis we can note things like: every closed subspace of a Banach space is Banach. In this case, what does "closed subspace" mean?

• Does this mean closed under the norm topology?
• Or does this mean closed in the sense that multiplication of scalars and addition of vectors is closed?
• Or does this mean closed with respect to limits?

I'm reviewing this material and I realized that even though I have this in my notes a number of times I am unsure of what this actually is. I thought it was the second statement above, but the third statement makes the "every closed subspace of a banach space is banach" statement easy to prove.

"Does this mean closed under the norm topology?"

Yes, that's what it means.

"Or does this mean closed in the sense that multiplication of scalars and addition of vectors is closed?"

No, that is what "subspace" means here. (As user1736 says, sometimes people write "linear subspace" for emphasis, but in functional analysis it is generally safe to assume that "subspace" means "linear subspace".)

"Or does this mean closed with respect to limits?"

Again yes, because this is an equivalent characterization of closed subspaces of a topological space. Since a Banach space is metrizable hence first countable, it is enough to take limits of sequences. For a general topological space -- and even some non-Banach topological vector spaces -- in order to retain this equivalence, one must allow limits of nets.

I think "subspace" usually refers to closure under scalar multiplication and vector addition, while "closed" refers to closure under the topology. Sometimes people like to say "linear subspace" instead of subspace to be more precise, but I've found that in a lot of functional analysis textbooks, the linear part is often implied.