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Why are vector spaces not isomorphic to their duals?
Dual space question

Can someone give an (as easy as possible) example (together with a proof) of an infinite dimensional vector space that is not isomorphic to its dual ?

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marked as duplicate by t.b., joriki, Hans Lundmark, anon, Jonas Teuwen Sep 8 '11 at 9:55

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We can consider $l^1$, the vector space of the sequences $\{x_n\}$ such that $\displaystyle\sum_{n=1}^{+\infty}|x_n|<\infty$. The dual can be identified with $l^{\infty}$, the vector space of all bounded sequences. But these two spaces cannot be isomorphic, since the first is separable whereas the second is not.

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