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.