Twenty Questions for the Final

1. (a) Let (X, ‖ · ‖) and (Y, ‖ · ‖) be normed spaces. Show that the following conditions on a family T ⊆ L(X, Y ) are logically equivalent: (1) {‖T ‖ : T ∈ T} is bounded in R; (2) {T : T ∈ T} is equicontinuous at some point of X ; (3) {T : T ∈ T} is equicontinuous at 0 ∈ X ; (4) {T : T ∈ T} is equicontinuous at every point of X . (This order may not be the most efficient one to prove round-robin implications.)