Q:

Definition of linear combination for an infinite subset $G$ of a vector space $$ over a field $K$.

My book offers two definitions of linear combination:

Let $<V,+,·>$ be a vector space over a field $K$, and $G = \{ v_1, ..., v_r \} ⊆ V$. A Linear Combination of $G$ is an element $v∈V$ such that $v=\sum_{i=1}^r α_i · v_i$ with $α_i∈K$ for every $1≤i≤r$.

I get this one, but then it offers a "more general" one that says:

Let $<V,+,·>$ be a vector space over a field $K$, $I$ a set of indexes and $G = \{v_i|i∈I\} ⊂ V$. A Linear Combination of $G$ is an element $v∈V$ such that $v=\sum_{i∈I} α_i · v_i$ where $α_i=0$, except for finite $i∈I$.

I don't get this one, and I think it's because I don't understand the role of the set $I$, also I have never used $\sum_{i∈I}α_i$ notation, just the regular one as in $\sum_{i=1}^r α_i · v_i$. In addition, I don't understand if when G is an infinite set then all $α_i$ have to be equal to $0$, or just some of them. 

A:

$\sum_{i∈I}α_i$ is defined naturally.  Let's say $I=\{\text{Alice},\text{Bob},\text{Charlie}\}$. Then $$\sum_{i∈I}α_i = \alpha_{\text{Alice}} + \alpha_{\text{Bob}} + \alpha_{\text{Charlie}}$$
The only important thing is that when $|I| = \infty$, you require that only finitely many $\alpha_i$'s can be non-zero because addition of an infinite number of vectors in a vector space is undefined (unless there's additional structure like a topology on it.  But for a vector space without any additional structure, it's undefined). 
For example, polynomials are generated by the set $\{1,x,x^2,\cdots,x^n,\cdots\}$. So, the set of polynomials consists of all linear combinations $c_o\cdot 1+ c_1\cdot x + c_2\cdot x^2 + \cdots$ such that except finitely many of the coefficients, the rest are zero.  So, the formal power series for $e^x$ is not a polynomial for example.