Recall that we saw that there was no natural isomorphism between a finite dimensional vector space, \(V\), and its dual, \(V^*\). When we have a non-degenerate inner product, the situation is a little different. Consider first the case of a real non-degenerate inner product space, and define a map \(V\mapto V^*\) according to \(v\mapsto f_v\), where \(f_v(w)=(v,w)\) for any \(w\in V\). That this is indeed a linear map follows since \(f_{av}(w)=(av,w)=a(v,w)\), that is, \(av\mapsto af_v\). It is injective since the inner product is non-degenerate, and, since \(\dim V=\dim V^*\), it is an isomorphism. In the complex case we again have a bijection but now, since \(f_{av}(w)=(av,w)=a^*(v,w)\) it is no longer linear but antilinear. Thus, in this case, we say that the map is an antiisomorphsim.