More generally speaking, an isomorphism is a bijection between two objects that preserves their structure. The meaning of structure depends on the category you're working in.
- Sets: An isomorphism of sets is just a bijective map.
- Topological spaces: An isomorphism is a bijective continuous map whose inverse is continuous (thus, preserves open sets).
- Groups: An isomorphism is a bijective map that's a homomorphism (thus, preserves the group operation).
- Vector spaces: An isomorphism is a bijective map that's a linear transformation (thus, preserves the linear structure).
The list goes on. In your case, I will add that many times a vector space also has a topology (such is the case with $\mathbb R^n$, for example). In this case you'd be interested in an isomorphism of topological vector spaces, that is, a bijective map that's linear, continuous, with continuous inverse. It turns out that some of these requirements are superfluous :)
As for the notation for isomorphic spaces - I don't know if there's a standard one. It seems every book has its own favorite version of the equality symbol.