From 'Functional Anaylsis' by Bachman
If the linear transformation is 1:1, it is called an isomorphism
Is this right? I thought an isomorphism was a morhpism that admitted an inverse. Unless we also know that this linear transformation is onto, how can we know that is has an inverse?
If a linear transformation is one-to-one, then it has some range $R$ which is likely a subspace of a Euclidean space $\Bbb{R}^n$. While it might not be onto with $\Bbb{R}^n$, it is definitely onto with $R$ and thus the linear transformation is an isomorphism between the domain and $R$.