Between two normed (linear) spaces there are several notions of isomorphisms:
- Linear isomorphisms: linear bijective maps
- Topological isomorphisms: linear homeomorphisms (due to the linearity these are automatically uniformly continuous and even bounded)
- Isometric isomorphisms: linear surjective isometries
Obviously isometric isomorphisms are also topological isomorphisms and topological isomorphisms are also linear isomorphisms and these inclusions are strict in the case of infinite-dimensional normed spaces. The advantages of topological isomorphisms are evident:
- they preserve all topological properties
- they preserve completeness
The advantages of isometric isomorphisms are less clear to me. What properties that are invariants of isometric isomorphisms but not under topological isomorphisms? Is there some kind of classification? The completeness might suggest that properties that can be purely expressed by the metric $d(x,y)=\lVert x-y\lVert$ are invariants under topological isomorphisms, but I'm not sure about that (Are uniformly homeomorphic metric spaces distinguishable?). One obvious example would be the "Hilbertness", namely the parallelogram law which is not preserved by topological isomorphisms. What would be other important examples?
Do you have any reference to a list which properties are invariants under which isomorphisms?