# How can I show that two infinite-dimensional vector spaces are isomorphic?

How can I show that two infinite-dimensional vector spaces are isomorphic?

Can I define a function which maps a basis of the vector space $V$ to a basis of the vector space $W$? And are there more ways to show that vector spaces are isomorphic?

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By explicitly exhibiting an isomorphism. Or by exhibiting monomorphisms in both directions. or by exhibiting epimorphisms in both diections. Or by determining the dimensions. –  Hagen von Eitzen Jul 21 '13 at 18:08
Note that not every two infinitely dimensional spaces are isomorphic, of course. It depends on the cardinality of the basis. Also note that there is no guarantee that you can write down a basis explicitly. –  Asaf Karagila Jul 21 '13 at 18:18
Perhaps it is worth to consider a special case of your question: when is an infinite-dimensional vector space isomorphic to its dual vector space, and how can one show that they isomorphic (or not). –  Dietrich Burde Jul 21 '13 at 20:57
If they are Hilbert Spaces you may simply check if they are separable or not. –  toypajme Jul 22 '13 at 5:09