If a Banach space is
isomorphic in the vector and norm sense
to another Banach space, does this imply the two are isometrically isomorphic?
I think so, because: from Wikipedia
- "an isometry [...] is a linear map f which preserves the norm"
- "a surjective isometry [...] is called an isometric isomorphism"
and Definition 1.3 from here
- an isomorphism between two vector spaces is a map that is one-to-one and onto and preserves structure.
So if two Banach spaces are isomorphic in both the vector sense and the norm sense, they must be isometrically isomorphic?