Question in regards to definition: finite dimensional Do we denote a vector space as finite dimensional IF it has a basis, or do we say that it is finite dimensional if it's associated through an isomorphic transformation with a "number space", ie. $\mathbb{R}^n$? 
Also, are all number spaces, $\mathbb{R}^n$ finite dimensional?
 A: A rigorous definition of a finite dimensional vector space might be:

A vector space is finite dimensional if there exists a finite number of vectors that span that space.

You might go on to prove that there is a well-defined minimum number of vectors that span a finite dimensional space, and we call this number the dimension of the space. In particular, any two bases for the same space have the same cardinality.
A: *

*Do we denote a vector space as finite dimensional IF it has a basis?


No. A vector space is finite-dimensional if it has a finite basis. An infinite dimensional vector space may (@Seth tells me must) have a basis. For example the polynomials in $x$ have a basis $1, x, x^2 \ldots \ $ .


*

*... or do we say that it is finite dimensional if it's associated
through an isomorphic transformation with a "number space", ie.
$\mathbb{R}^n$?


No. There are many vector spaces that have nothing to do with numbers. Even within numbers, you can restrict reals to rationals and you still have a vector space.


*

*Also, are all number spaces, $\mathbb{R}^n$ finite dimensional?


Yes. They have, as @Seth says, dimension $n$. 
