TITLE: Linear algebra doubt about the use of the word 'finite'
QUESTION [12 upvotes]: Let $V$ be a vector space and $S$ be a non empty subset of $V$. If $v\in\operatorname{Span}(S)$ then it is a linear combination of a 'finite' number of vectors in the subset $S$. What does the word finite signify? Could have just said the linear combination of the vectors of $S$.

REPLY [3 votes]: It's worth pointing out that there can be vector spaces where some "infinite linear combinations" would make perfect sense. 
Consider the vector space $\mathbb{R}^{\omega}$ without the restriction of finite support. This is a perfectly good vector space over $\mathbb{R}$ if you define addition coordinate-wise and multiplication by scalar you get a standard vector space. 
In this vector space if you have any countable set of vectors $v_i$ and scalars $a_i$ that satisfy the condition that $\{i|\;j\in \mathrm{supp}(v_i)\}$ is finite for each $j$, then the $\sum_0^\infty a_i v_i$ is quite well defined and you could easily consider calling this a linear combination. 
This is another reason to specifically call to attention the finiteness of the linear combination even though in most courses that is part of the definition.