What is the difference between n-dim ($n<\infty$) Banach space and $R^n$?

Question

What is the difference between n-dim ($n<\infty$) Banach space and $R^n$?

I feel they are same ,no matter under homeomorphism isomorphism or diffeomorphism. In fact, I feel they are not different.

Answer 1

A Banach space is a vector space together with a choice of a (complete) norm for that space. So, in fact, $\mathbb R^2$ has many different "Banach space" structures. Indeed, as you note, they are all homeomorphic to each other. But they are not isometric.

Answer 2

It is true in a technical sense. But that doesn't mean that the concrete form of the norm is irrelevant for all uses.

Even in infinite dimension, for instance if you are considering Hilbert spaces, $\ell^2 (\mathbb N) $, $\ell^2(\mathbb Z) $, $L^2(\mathbb T)$, and $L^2 (\mathbb R) $ are indistinguishable. Still, you deal with the unilateral shift in the first one, with the bilateral shift in the second one, with Fourier series in the third one, with $C_0 (\mathbb R) $ in the fourth one. Etc.