1. Prove that a Banach space X is reflexive if and only if its dual X 1 is reflexive. Note: "Only if" is easier.
2. Let (X, 1·1) be a normed linear space, E a closed linear subspace, and x ∈ X \E . Prove that for some f ∈ X t, f ≡ 0 on E and f (x ) = 1.
3. Show that the extension theorem for Lipschitz functions (6.1.1) does not hold for complex-valued functions. Hint: Let S = R2 with the norm 1(x, y)1 := max(|x |, |y|). Let E := {(0, 0), (0, 2), (2, 0)}, with f (0, 0) = 0, f (2, 0) = 2, and f (0, 2) = 1 + 31/2i . How to extend f to (1, 1)?