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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?

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  • $\begingroup$ 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 $n$. $\endgroup$ – Seth Dec 15 '14 at 16:20
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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.

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  • 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 $n$.

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

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