# Finite Dimensional Inner-Product Space Equals its Dual?

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!

Fredrik
Staff Emeritus
Gold Member

They are isomorphic, not equal. Hence the phrase "essentially equal".

You only need the Riesz representation theorem when V is infinite-dimensional, since you can easily see that any two n-dimensional vector spaces (with the same n) are isomorphic.

The isomorphism between V and V** is more interesting, because it can be defined without an inner product or a choice of bases for V and V**. The function f:V→V** defined by f(x)(ω)=ω(x) for all ω in V* and all x in V is such an isomorphism.

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