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.