A set S consists of those vectors with a finite number of nonzero components.
Given a vector space V with infinite dimensions, or $V = R^\infty$, I am trying to prove that the set S is a subspace of it.
This is Problem 18 in Section 7.1 (Vector Spaces and Subspaces), page 285, from Linear Algebra by Jeffrey Holt.
By definition, to be a subspace, S must:
- Contain the zero vector
- Be closed under vector addition
- Be closed under scalar multiplication
I immediately want to say that this proof is false because of the first point above with the zero vector. Since S consists of those vectors with a finite number of nonzero components, it makes me think that it cannot contain the zero vector which would have the same number of components as S, with all of them being 0 (e.g. The zero vector contains a finite number of zero components!)
Can someone help me understand this better? Is my logic correct?