# Vector isomorphisms, Manifolds and Lie groups

This is an excerpt of Lee's book on smooth manifolds.

As far as I understand, if there is a (vector/linear) isomorphism between two vector spaces, then they are indistinguishable in the eyes of "vector mathematics", meaning that any proposition whose hypotheses contain nothing else but concepts defined in terms of the axioms for vector spaces vector space operations is either true for both vector spaces or false for both vector spaces.

Now: two isomorphic vector spaces which are also manifolds may not be indistinguishable from one another in the eyes of "differential mathematics", because their differential structure may be different. That said, I do not understand why Lee writes that, since $GL(V)$ and $GL(n,\mathbb{R})$ and the latter is a Lie group, then $GL(V)$ is also a Lie group. After all, the definition of Lie group contains concepts which are defined outside of the so called "vector mathematics".

• With an atlas of $GL(n)$ and that isomorphism you have an atlas of $GL(V)$. Now you just need to check that multiplication is smooth. But this is the case, because of the above atlas. – T'x May 8 '17 at 19:52

Any vector space isomorphism from $V$ to $\mathbb{R}^n$ is also smooth. This is because linear maps are smooth.

I think the reason for your confusion is that your vague definition of vector space isomorphism is not correct. I suggest reviewing the definition of a linear transformation and isomorphism, and then rethinking your explanation of isomorphism. Any proprosition which can be proved using the vector space axioms is true for all vector spaces, but not all vector spaces are isomorphic. Any two vector spaces of the same dimension are isomorphic, and you can use this to refine your statement to something true ... however, an isomorphism is an explicit transformation witnessing the equivalence (which yes, allows you to transfer structure), and it is important to understand this. You should know why a basis for a vector space $V$ (finite dimensional) determines a linear isomorphism with $\mathbb{R}^n$.

Understanding linear algebra on this level is definately a pre-requisite for learning manifold theory.