MATHEMATICS 421/510, PROBLEM SET 2 Due on Thursday, February 2

Due on Thursday, February 2
Write clearly and legibly, in complete sentences. You may discuss the
homework with other students, but the final write-up must be your own. If
your solution uses any results not introduced in class, state the result clearly
and provide either a reference or a proof.
1. (10 points) Let X be a linear space over R, and let K ⊂ X be a convex
set. An extreme set of K is a non-empty and convex set E ⊂ K such
that if x ∈ E can be written as x = (y + z)/2 with x, y ∈ K, then both
x and y belong to E.
Prove that if K has an interior point and E is an extreme set of K such
that E 6= K, then there is a linear functional ` : X → R, ` 6≡ 0, and a
constant c ∈ R such that
sup `(x) = inf `(x) = c.
2. (10 points) What is the completion of the space of simple functions on
S = {f =
ci χEi : Ei ⊂ R measurable , |{x : f (x) 6= 0}| < ∞}
in the L∞ norm? (Hint: follow the proof that C0 (R) is the L∞ completion of Cc (R), except that of course you should expect to get a different
3. (10 points) Prove that the k · kp norm on Lp (R), 1 ≤ p ≤ ∞, does not
satisfy the parallelogram identity
kf + gk2p + kf − gk2p = 2kf k2p + 2kgk2p
if p 6= 2. (Hence Lp (R) is not a Hilbert space for p 6= 2.)
4. (Conway, Example 1.8; 10 points) Let H be the collection of all absolutely continuous functions f : [0, 1] → C such that f (0) = 0 and
f 0 ∈ L2 (0,
R 11).0 Prove that H is a Hilbert space with the inner product
(f, g) = 0 f (x)g 0 (x)dx.