How to prove that $F^{\infty}$ is a vector space I am working with the space that Axler defines as 
$$F^{\infty} := \{(x_1, x_2, \ldots, ) : x_j \in F\}.$$
This is a vector space over $F$, which Axler asks the reader to verify. I didn't have any trouble verifying that $F^n$ is a vector space: it isn't tough to verify that addition and scalar multiplication are defined, addition is commutative and associative, scalar multiplication is associative, there exist inverses, and so forth. $F^{\infty}$ is by definition a set of lists, but these lists are intended to be finite by definition, so it sounds rather peculiar to have a countably infinite list, meaning that the usual addition of 
$$(x_1, x_2, \ldots, x_n) + (y_1, y_2, \ldots, y_n) = (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n)$$
isn't defined. Am I correct about this, or can we extend it to the infinite case? 
I can see three ways to verify this: 
(1) Extending the definition of "list" to the countable case, wherein all of the properties follow from the properties of $F$, as in the $F^n$ case. 
(2) Prove the result for an arbitrary entry ,$i$, of $F^{\infty}$, in which case it holds for all entries. 
(3) $F^{\infty}$ represents the set of functions from $\mathbb{N}$ to $F$. In the same way that we can prove that $F^S$, the set of functions from a nonempty set $S$ to $F$ is a vector space, we can show that this set of functions is a field, and hence $F^{\infty}$ is a vector space. 
Which of these is the "correct" way to go about it?
 A: I'm maybe a little confused about why you're confused. But maybe some of what I'll say next will help.
(1) and (3) are basically the same. There's no difference between $(x_1, x_2, \dots)$ and $f(i) = x_i$. One could argue that they are different encodings meaning that they are literally different. However, in terms of what properties these two encodings have and what information they contain, there is no difference. The properties matter, the encoding does not.
(2) and (3) are basically the same as well just because the statement "$f(i) = x_i$" could mean either: for a fixed coordinate $i$, look at $f(i)$ and that is equal to $x_i$; or it could mean $f$ is the function that maps an arbitrary input $i$ to $x_i$. The difference being whether or not you are considering $f(i)$ as being the function $f$ or $f(i)$ as being the specific value of that function at $i$.
So I suppose in summary, there's nothing wrong with talking about an infinite list (also known as a sequence). Like you're probably comfortable with the Fibbonacci sequence: $F_0 = 0$, $F_1 = 1$, $F_n = F_{n-1} + F_{n - 2}$. This is an infinite sequence. It doesn't matter if I write it as $F_n$ or $F(n)$ that's just notation. All of (1), (2), (3) are correct ways to think of infinite sequences and it is often good to think of objects in multiple ways.
