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 
Senior Member Joined: Aug 2012 Posts: 1,373 Thanks: 322  

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dual, space, vector 
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