inner product space: = N sum f(x)g(x) bar (g(x) bar - conjugate)

4. Let G be an ordered graph, i.e., a pair (V, E) where V is a finite set of vertices and E CV XV is a set of edges. Let L(V) and L(E) denote the vector spaces of complex-valued functions on V and E respectively, endowed with the natural inner product space structure described in class. Let D: L(V) + L(E) be the linear transformation defined by (Df)((V1, V2)) = f(v2) f(vi), for all (V1, V2) E. Describe the adjoint D* of D, and compute D*D. 4. Let G be an ordered graph, i.e., a pair (V, E) where V is a finite set of vertices and E CV XV is a set of edges. Let L(V) and L(E) denote the vector spaces of complex-valued functions on V and E respectively, endowed with the natural inner product space structure described in class. Let D: L(V) + L(E) be the linear transformation defined by (Df)((V1, V2)) = f(v2) f(vi), for all (V1, V2) E. Describe the adjoint D* of D, and compute D*D