How to define a vector without a basis? We know that a vector is an element of a vector space that does not depend by the basis that is chosen to represent it. But, in a finite dimensional vector space, when we want define a specific vector, we usually gives its components with respect to some basis. Than we can change the basis and the components of the vector change in a suitable manner such that the vector remain, geometrically, the same.
But I dont know a simple way to define a vector without give his components with respect to some basis.
The situation seems different in an infinite dimensional vector space, as a space of function. In this case ve can define a vector as a function that satisfies some conditions ( e.g. n $L^2$ function), and this does not require a basis, so, doing this, we have e definition of the vector (function) that does depend from a basis.
If we want, we can now use a basis and define the components of the given function as coefficients of a series development ( a generalized Fourier expansion) and these coefficients are the analogous of the components of a vector in a finite dimensional space.
So it seems that in an infinite dimensional vector space we can define the vectors in an ''intrinsic'' way, without using a basis, but this is not possible (really ?) in a finite dimensional space.
It's this really true? and there is some motivation?

Finite Example
Question: How do you define $\mathbb{R}^N$?
Sloppy answer: $x=\sum_{n=1}^Nx_ne_n$
Correct answer: $x=(x_1,\ldots,x_N)$
Formal answer: $x:\{1,\ldots,N\}\to\mathbb{R}$
(That is your basis-free representation.)
Infinite Example
Question: How do you define $\ell^2(S)$?
Answer: $x:S\to\mathbb{C}:\quad\ldots$
(That is the analogue to the above.)
