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.
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