Background
I know this question might be a bit weird. More out of curosity.
I know from the definition that one of the criteria for $V$ to be a vector space is that $\forall v_1,v_2 \in V, v_1 + v_2 \in V$.
But how about countable infinite sum? From my viewpoint, it sounds more like a vector space when infinite countable sum is still contained in $V$.
Example
I have this question because when I look at the $l^{\infty}$, i.e., the bounded sequence space. I feel like if I can recursively sum up two proper sequence in $l^{\infty}$, I can make resulting sequence larger than any given bound. So it sounds like, it might be out of the vector space that I am interested in.
I know the definition of vector space requires "two element sum still contained in the space". Just curious if there is a reason for not allowing infinite sum.