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**Finite Dimensional Inner-Product Space Equals its Dual!?**

Let V be a finite dimensional inner-product space. Then V is 'essentially' equal to its dual space V'.

By the Reisz Representation theorem, V is isomorphic to V'. However, I've been told that V=V', which I am having a hard time believing. It seems to me that the two spaces do not contain the same elements: V' contains linear functionals, while V contains any kind of vectors. Therefore, V does not equal V'.

Could someone clear this up for me? Is V only isomorphic to V', or are the two spaces REALLY equal?

Thanks!