1. The problem statement, all variables and given/known data T ∈ L(V,W). Thread title. 2. Relevant equations 3. The attempt at a solution Note that T* is the adjoint operator. But there's one thing that I need to get out of the way before I even start the proof. Now consider <Tv, w>=<v, T*w> w in W, v in V. Now when they say T is injective if and only if T* is surjective, <Tv, w>=<v, T*w> only makes sense when W=V since Tv is supposed to map to W and if dim W<dim V, T is not injective.