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 March 17th, 2017, 08:27 AM #1 Senior Member   Joined: Aug 2014 From: United States Posts: 136 Thanks: 21 Math Focus: Learning dual vector space I know that $(V^*)^*\cong_{vec} V$ for finite dimensional vector spaces $V$. However, it also seems as if $V\cong_{vec} V^*$, since for finite dimension, $V$ and $V^*$ are both vector spaces of the same dimension, and any two vector spaces of the same finite dimension are isomorphic. But I know that $V$ and $V^*$ are not isomorphic as vector spaces, so what's wrong with my above logic?
March 17th, 2017, 10:09 AM   #2
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 Originally Posted by neelmodi But I know that $V$ and $V^*$ are not isomorphic as vector spaces, so what's wrong with my above logic?

They are isomorphic, but not "naturally," because you have to pick a particular basis to show the isomorphism.

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