There is a theorem in linear algebra that if two vector space are of same finite dimension over the same field then they are isomorphic to each other. Now my question is that if two vector space are of same infinite dimension over the same field are they are isomorphic? If this result is true for infinite case then please suggest me how to prove it. If bases have same cardinality of two vectors spaces then there is one to one correspondence between their bases. But this one to one correspondence give results in finite cases. For infinite case i am stuck. Please suggest me. Thanks in advance.

  • $\begingroup$ "One-to-one correspondence" means "bijection", there is no dependence on the sets being finite to conclude anything. $\endgroup$ – Zev Chonoles Jul 17 '15 at 5:08
  • $\begingroup$ If you can assume that the infinite-dimensional vector spaces have bases, then as Zev Chonoles said in the comment to an answer, a bijection between basis elements gives you an isomorphism (the proof is very similar to the finite-dimensional case). The complication is that the existence of a basis for an arbitrary infinite-dimensional vector space depends on the axiom of choice. See here: math.stackexchange.com/questions/86762/… and here: mat.uniroma2.it/~tovena/infinite.pdf $\endgroup$ – coldnumber Jul 17 '15 at 5:27

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