# Isomorphisms between Normed Spaces

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?

• Pedantic comment: There are no linear isomorphisms between normed spaces. What you really do is to apply the forgetful functor from normed spaces to vector spaces and consider isomorphisms in that category. Always distinguish between an object and its image under some forgetful functor. Vector spaces are boring, they are just direct sums of copies of the base field. This is not true for normed spaces. – Martin Brandenburg Dec 29 '13 at 16:05
• Idea to answer your question: What about some kind of volume of the unit ball? – Martin Brandenburg Dec 29 '13 at 16:07

Isometric isomorphisms preserve:

1) Hilbertness. This can show that isometric image of Hilbert space is also Hilbert. But Hilbert space is so nice and sweet that even after applying topological isomorphism it is still recognizible.

2) Extreme points. This fact is often used to prove non-existence of isometric isomorphism between two normed spaces.

3) Group of isometries. Knowledge about this group sometimes give complete description of other mathematical structure used to construct a Banach space. See for example Banach-Stone theorem. By the way topological isomorphism can completely destroy the group of isometries of any Banach space.

4) $1$-complementabilty. This is the best possible way a normed space can sit inside ambient normed space, and this property is preserved under isometric isomorphisms. For example $X^*$ is always $1$-complemented in $X^{***}$

5) Gateaux and Frechet differentiabilty. This property can be used to prove non-existence of isometry between $\ell_\infty$ and $L_\infty$

6) Metric characteristics. There are a lot of them. I'll just mention basic constants of basic sequences. Basic sequences play a crucial role in the study of geometry of Banach spaces. Basic constant is real number showing how far a sequence of vector from "being orthogonal". Under isometric isomorphism this constant doesn't change.

See similar question here.