Comment by a57721

A vector is always a vector -- an element of something that satisfies the axioms of a vector space. The author starts with the example of R^n, which is a very particular vector space that is finite-dimensional and comes with a "canonical" basis (0,...,1,...,0). In general, a basis will always exist for any vector space (using the axiom of choice), but there is no need to fix it, unless you do some calculations. The analogy with R^n is the only reason the "indices" are mentioned, and I think this only creates more confusion.

> and they aren’t irrational (i.e. they have a finite precision)

No, if you want only rational "indices", then your vector space has a countable basis. Interesting vector spaces in analysis are uncountably infinite dimensional. (And for this reason the usual notion of a basis is not very useful in this context.)