Why vector space only requires a finite sum?

Question

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.

Answer 1

Consider the simple vector space $\mathbb Q$ (rational numbers). It is indeed the case that $$\forall q_1, q_2 \in \mathbb Q : q_1+q_2 \in \mathbb Q.$$

However,

$$1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \frac{1}{120} + \cdots + \frac{1}{n!} + \cdots = e \notin \mathbb Q.$$

So, infinite sums are effectively quite a different beast than finite sums as far as vector space closure is concered.

Answer 2

Think of $\Bbb R$, which we all know and love. It is a fine vector space. We know how to add any two reals, which tells us how to add any finite collection of reals. We can't use that to define the sum of an infinite sequence of reals. We go through all sorts of machinations to define the sum of some infinite sequences of reals, which we call convergent. The rest we can't give meaning to. What would you like the sum of $1+1+1+1+\ldots $ to be?