= 0. (Here A* denotes the adjoint of A and Ker(A*) denotes the kernel = null-space of A*.) In your proof, do you need to assume anywhere that either of V or W is a finite dimensional vector space? Can you do without such an assumption? Do you assume any other property of A? Investigate how the Fredholm Alternative theorem is used in the proof of Range(L) = Range (LL*) - used in class ( see also the Appendix A of Tits lecture notes). 2. In the capacitor charging problem in the class notes (see page 14 of Tits lecture notes), put a linear inductance L in series. Then solve the problem of charging. State clearly any asumptions you make along the way. What is the nature of the optimal charging current? Does the resonance frequency play a role in your answer? What is the efficiency? Compare your answer with the case of the inductance-free circuit. What is the form of the answer if you are allowed arbitrarily long charging times?