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?

• All vector spaces have a basis. A space is finite dimensional if it's basis is finite. Yes, all $\mathbb{R}^n$ have finite dimension, specifically dimension $n$. – Seth Dec 15 '14 at 16:20

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.

• 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 $n$.

• One might add that if a real vector space is finite dimensional, then it is isomorphic to $\mathbb{R}^n$ for some $n$. – Thomas Dec 15 '14 at 17:28
• @Thomas One might indeed. – Thumbnail Dec 15 '14 at 17:30