Mathematically speaking, if I were to work on an arbitrary real vector space $E, (dim E = n)$, would proving theorems on $\mathbb{R}^n$ make them valid in $E$ ?
We know that if we fix our field, in this case $\mathbb{R}$, any two $\mathbb{R}$-space, vector space, of dimension $n$ would be isomorphic to each other, so if I were to prove some theorem valid on an arbitrary vector space $E$ defined on the field $\mathbb{R}$, can I just treat $E$ as $\mathbb{R}^n$ and make the proof and say that this is proof also valid on $E$ ?
If so, while doing the proof on $\mathbb{R}^n$, If I treat $\mathbb{R}^n$ as a model for $n$ dimensional geometry, would this make any difference for the case above ?