MATHEMATICS 421/510, PROBLEM SET 2
Due on Wednesday, February 3
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. All page and section numbers below
refer to the textbook, “Functional Analysis” by Peter D. Lax.
1. (a) Prove that L∞ (R) is not separable.
(b) What is the completion of the space of simple functions on R,
S = {f =
n
X
ci χEi : Ei ⊂ R measurable , |{x : f (x) 6= 0}| < ∞}
i=1
∞
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
space.)
2. (a) Prove that the norms on L1 (R) and L∞ (R) are not strictly subadditive. (See p. 44 for the definitions.)
(b) Prove that if a normed space is uniformly convex then its norm must
be strictly subadditive. (Hint: Prove first that it suffices to consider
the case when |x| = 1, |y| > 1. Let y 0 = y/|y|, then |x − y| ≤ |x − y 0 | +
|y − y 0 |.)
3. (Exercises 1 and 2, page 53) (a) Show that a norm that satisfies the
parallelogram identity comes from a scalar product.
(b) Show that the scalar product depends continuously on its factors;
that is, if kxn − xk → 0, kyn − yk → 0, then (xn , yn ) → (x, y).
4. (a) (Exercise 3, page 39) Let X be a Banach space, Y closed subspace
of X, and define a norm on the quotient space X/Y as on page 37.
Prove that X/Y is complete in this norm.
(b) Let Y be a closed subspace of a Hilbert space X. Define a linear
mapping T from the orthogonal complement Y ⊥ to the quotient space
X/Y via T x = [x], the equivalence class of x in X/Y . Prove that T
is one-to-one, onto, and that |T x| = kxk, where the norm on X/Y is
defined as in (a).
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