1. Show that the closed unit balls of c0 and L1 [0, 1] have no extreme points. 2. Let T be a homeomorphism of a compact metric space K. Recall that extremal points of the set of T -invariant probability measures are called ergodic measures. If µ is an ergodic measure then we say that T acts ergodically on (K, µ). (i) Show that a probability measure µ is ergodic if and only if there are no nontrivial invariant measurable subsets. That is, if A ⊂ K is µ-measurable and T A = A then either µ(A) = 0 or µ(A) = 1. (ii) Let K = T, T z = e2πiθ z. Show that if θ is irrational then the Lebesgue measure is ergodic. Hint: look at the Fourier series of the characteristic function of an invariant set. 3. A normed space is called strictly convex if the equality kx + yk = kxk + kyk is possible only when x = λy for some λ > 0 (assuming x, y 6= 0). Show that for any measure space (X, µ) and 1 < p < ∞ the space Lp (X, µ) is strictly convex. Remember though that in fact the Lp -spaces are even uniformly convex. 4. A normed space is called uniformly convex if for any ε > 0 there exists δ > 0 such that if kxk = kyk = 1 and kx + yk > 2 − δ then kx − yk < ε. (i) Show that any uniformly convex space is strictly convex. (ii) Show that if A is a closed convex subset of a uniformly convex Banach space X then for any point x ∈ X there exists a unique point of A that is closest to x. 5. Show that if K is a convex compact set in Rn then any point of K is a convex combination of at most n + 1 extremal points. Argue as follows. Let x ∈ K. Choose an extremal point x0 ∈ K. The intersection of K with the line through x and x0 is a segment of the form [x0 , y]. Show that there exists a hyperplane H passing through y such that K lies on one side of H. Continue by induction on the dimension with K ⊂ Rn replaced by (K ∩ H) ⊂ H. 1