1. In any infinite-dimensional F -space, show that any compact set has empty interior.
2. A continuous linear operator T from one Banach space X into another one Y is called compact iff T {x ∈ X : lx l≤ 1} has compact closure. Show that if X and Y are infinite-dimensional, then T cannot be onto Y . Hint: Apply Problem 10.
3. Show that there is a separable Banach space X and a compact linear operator T from X into itself such that T {x : lx l ≤ 1} is not compact. Hint: Let X = c0, the space of all sequences converging to 0, with norm l{xn }l := supn |xn |.