14. Prove Theorem 6.19 (Hellinger-Toeplitz Theorem). Let (X,(,), || ||) be a Hilbert space and A: XX be a linear operator. If A is self-adjoint, i. e., Vx,y X (Ax, y) = (x, Ay), then A € L(X). Hint. Use the Sequential Characterization of Closed Linear Operators (Proposi- tion 5.3) and the Inner Product Separation Property (Proposition 4.14) to show that A is closed and apply the Closed Graph Theorem (Theorem 6.16). 15. Prove Proposition 6.18 (Characterization of Orthogonal Projections). A projection oper- ator P on a Hilbert space (X, (, ), || I) is an orthogonal projection iff P is self-adjoint, i. e., Vx,y X (Px, y) = (x, Py). 16. Prove Proposition 6.19 (Characterization of Orthogonal Projections). A nontrivial pro- jection operator P on a Hilbert space (X, (,), || ||) is an orthogonal projection iff ||P|| = 1. Hint. To prove the "if" part, reason by contrapositive using the Characterization of the Orthogonal Complement of a Subspace (Proposition 4.6).