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.