isomorphisms of two vector spaces and their duals If two vector spaces are linearly isomorphic, then so are their duals. Is the converse true?
Thanks for your mention.
 A: If we're talking about algebraic duals, then the converse implication is at least possibly true, since it is implied by the Generalized Continuum Hypothesis, which is consistent with ZFC.
Under GCH, if the dimension of $V$ is $\aleph_\alpha$, then the dimension of its algebraic dual is $2^{\aleph_\alpha}=\aleph_{\alpha+1}$. This means that if $V^*$ and $W^*$ are isomorphic (i.e. have the same dimension), then so are $V$ and $W$.
Now you may not believe in GCH, but this argument at least shows that you cannot hope to find an explicit counterexample (which ZFC proves is a counterexample).
A: If a vector space is finitely generated it is isomorphic to it's dual space, so in this case the converse is true (you have a commuting square of isomorphisms).
I would venture to say that, if a vector space $V$ is not finitely generated, this might also be true, since for any basis of $V$ the dual basis spans the weak dual space of $V$ which is a subspace of $V$'s dual space. So certainly if two vector spaces' weak dual spaces are isomorphic, it should be true. I am just not sure if one can claim that a linear isomorphism would preserve weak dual spaces? The linear isomorphism will preserve the weak dual space as a subspace, but I'm not sure if it will actually be the weak dual space of the second vector space...maybe someone else can shed light on that. ok - I believe Henning Makholm has the correct answer for the infinite case.
