Can someone prove this to me? I know that if you have a finite dimensional vector space V with a dual space V*, then every ordered basis for V* is the dual basis for some basis for V (this follows from a theorem). But if you're just given an arbitrary vector space V. Let's say the Space of R^n over the field R, then how could the elements of that space be linear functions of another space (and thus V is the dual space of another space)? And what would that space be? All i can guess is that V can be thought of as set of constant functions...(adsbygoogle = window.adsbygoogle || []).push({});

The two main theorems im supplied with in this section are

Thm: Suppose V is a finite dimensional vector space with an ordered basis B = {x_1,...,x_n}. Let f_i (1<= i<=n) be the ith coordinate function wrt B, and let B* = {f_1,...,f_n}. Then B* is an ordered basis for V*, and, for any f element of V*, we have f = Sum i=1 to n[f(x_i)f_i

Thm: Let V be a finite dimensional vector space, and and define U: V->V** by U(x) = x' where x': V* -> F is defined by x'(f) = f(x) (F is a field). Then U is an isomorphism.

So again, I would like a proof of the statement "Every vector space is a dual of some other vector space" and an example using lets say R^n as the dual space. Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Every vector space is the dual of some other vector space

**Physics Forums | Science Articles, Homework Help, Discussion**