How to see a function as a vector in a vector space I know that strictly speaking my question is some sort of a duplicate of at least this previous one and I am quite sorry for that (usually I try to get the best from previous questions), but still I did not find an answer that suited my intuition, probably because not enough developed. Still, I would really like to grasp in the proper way this important point.   
As a side problem, I think that a question that general can be problematic to answer, so I see it is not easy. Anyway, coming up with a specific example (or minimal working problem) could be a decent starting point. Thus, here we are.
Let's take as a building block a continuous function over $[0,1]$. When we are asked to think about this object as a vector in a vector space, is this object a vector with infinitely many elements? 
Does it make sense what I wrote?
Thanks a lot to any feedback.
 A: There are two thought-patterns which I would like to encourage:


*

*Relax your mind and stop thinking of vectors as lists of numbers! Vectors are just elements of a vector space, and that just means you can add and subtract elements since they form an additive abelian group, and you can also multiply these things with scalars. (Of course I am omitting a few connecting axioms in the interest of brevity.) Functions can be elements of sets that are vector spaces, so there is no reason to think of them any differently.

*After digesting the last point, I'll tell you that you can still visualize function as vectors. It's just that this vector has a lot of entries, so many that you can't even write the entries sequentially. For each number $r\in [0,1]$, you can just think of the "$r$th coordinate" of the vector, and there is a real number there. The number at the $r$th coordinate is $f(r)$. In this way, this very long vector (with as many entries as $[0,1]$ has elements) captures the entire function. Finally, notice that vector addition and scaling  of these "function-vectors" matches the operation of pointwise addition and scaling of functions.
A: Your example is formally denoted as $C[0,1]$ and wr.t. pointwise addition and multiplication it forms a vector space (simply because it satisfies all the criteria for it). Now any element of a vector space is a vector in that space so any $f:[0,1]\longrightarrow \mathbb{R}$ which is continuous is a vector in $C[0,1]$. Now how many such $f$'s can you define ? obviously infinitely many of them. Hence $C[0,1]$ has infinite number of vectors.
