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I would like to collect a list of things that are preserved under isomorphisms and isometric isomorphisms. The reason is that I hope to get a better perception of their importance in Functional Analysis.

Definition: $(X,||.||)$ and $(Y,||.||)$ be normed spaces, then a continuous linear bijective map whose inverse is also continuous is called an isomorphism between them. If this isomorphism is an isometry we call it an isometric isomorphism.

I have three things so far: separability, completeness and Hilbertness(although I am more interested in general Banach spaces) are preserved by isomorphism. What are they also good for and what for do we need isometric isomorphisms?

The best answer would be one with many properties that are preserved by isomorphisms and refers to things that somebody would know who has attended a first course on functional analysis.

Since most answers kept on saying, that "all" properties are preserved, I will list a few and maybe you can tell me whether they are preserved:

In the following I will always refer to something is mapped onto something with the same property by an isomorphism :

Reflexive spaces, closed subspaces, dense sets, linear independent vectors, compact sets, open sets, closed sets, disjoint sets.

If $X,Y$ are isomorphic, then also $X',Y'$.

These were a few properties. Is it correct, that they are always preserved?-But still, I would highly appreciate it if anybody could add a few things more. Probably some of them are also preserved by jut continuous (and injective/surjective) maps, would be interesting to know which ones.

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  • $\begingroup$ linear isomorphisms? $\endgroup$ – C-Star-W-Star Jan 13 '14 at 23:19
  • $\begingroup$ oh yes, sure... $\endgroup$ – user66906 Jan 13 '14 at 23:21
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    $\begingroup$ Well, then basically every property that is related to normed spaces??? I mean, two spaces being isomorphic means their indistinguishable except having different names??? $\endgroup$ – C-Star-W-Star Jan 13 '14 at 23:23
  • $\begingroup$ unless you are in category Ban all properties formulated in terms of this category are preserved by isometric isomorphisms. But if add weak topologies things some properties could be lost under isometric isomorphisms. $\endgroup$ – Norbert Jan 13 '14 at 23:33
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    $\begingroup$ @Freeze_S, I mean weak* topologies. In 2012 M. Daws constructed a countinuum of pairwise isomorphic preduals of $\ell_1$ with which are generate pairwise different weak* topologies on $\ell_1$. $\endgroup$ – Norbert Jan 18 '14 at 19:49
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Here are two answers to exactly the same questions asked earlier:

Isomorphisms between Normed Spaces

Why isometric isomorphic between Banach spaces means we can identify them?

If you are highly interested in isometries between Banach spaces, then I also recommend The isometric theory of classical Banach spaces. H. E. Lacey. It contains more advanced parts of Banach space theory.

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Every topological property is preserved. Unless it being uniformly continuous properties purely related to metric spaces are not preserved, e.g. being cauchy will be no longer preserved. There is another remarkably property that won't be preserved by uniformly continuous namely the potential to be a C*-Algebra rather than just a Banach Algebra. This is important in understanding spectral theory.

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  • $\begingroup$ I appreciate your answer, although I still wait for more extensive answers. $\endgroup$ – user66906 Jan 14 '14 at 0:27

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