Let V be a finite-dimensional vector space over F. The vector space V* = L(V,F) is...

    

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Let V be a finite-dimensional vector space over F. The vector space V* = L(V,F) is called the dual of V. An element of V is thus a linear map f: V F, also known as a functional. (i) Let (,...,Un} be a basis of V. Convince yourself that, for each i = {1,...,n}, there exists a unique V satisfying v (v ) = dij. Show that {u,..., v} is a basis of V. Deduce that V and V* are isomorphic. (ii) For each v V, let ev(v): VF denote the evaluation at v; that is, ev(v)(f) = f(v) for each f V. Check that ev(v) V, the dual of V*. Show that the evaluation map ev: V V** is an isomorphism. Let V be a finite-dimensional vector space over F. The vector space V* = L(V,F) is called the dual of V. An element of V is thus a linear map f: V F, also known as a functional. (i) Let (,...,Un} be a basis of V. Convince yourself that, for each i = {1,...,n}, there exists a unique V satisfying v (v ) = dij. Show that {u,..., v} is a basis of V. Deduce that V and V* are isomorphic. (ii) For each v V, let ev(v): VF denote the evaluation at v; that is, ev(v)(f) = f(v) for each f V. Check that ev(v) V, the dual of V*. Show that the evaluation map ev: V V** is an isomorphism.