MATH 421/510
HW 4
Due: March 22, 2016 (in class)
1. Let X be a Banach space and let T : X → X ∗ be a linear map (not
necessarily bounded) satisfying
(T (x))(y) = (T (y))(x)
for all x, y ∈ X . Show that T is a bounded operator. (Hint: Use the
uniform boundedness principle).
2. P
Let α = (αn ) be a given sequence of real numbers. Assume that
3
3/2
.
n |αn | |xn | < ∞ for every element x = (xn ) in ` . Prove that α ∈ `
(Hint: Use the uniform boundedness principle on an appropriate sequence of operators. Also note that the converse of the above statement
follows from Hölder inequality).
3. Suppose that (X, µ) is a measure space such that µ(X) < ∞. Let
T : L2 (µ) → L2 (µ) be a bounded operator. Suppose that the range of
T is contained in L3 (µ). Show that T is bounded as an operator from
L2 (µ) into L3 (µ). (Hint: Use the closed graph theorem).
4. Let X and let Y be normed vector spaces and let T ∈ L(X , Y). Recall
that the adjoint T ∗ ∈ L(Y ∗ , X ∗ ) is defined by T ∗ f = f ◦ T for all
f ∈ Y ∗ . Show that T ∗ is injective if and only if the range of T is dense
in Y.
5. Let X and Y be two Banach spaces and let T ∈ L(X , Y). In the class,
we saw that every finite-rank operator is compact. In this exercise, we
prove a partial converse. If T is a compact operator and the range of
T is closed in Y, show that T is a finite-rank operator. (Hint: Use the
open mapping theorem).
6. Let X = `p with 1 ≤ p ≤ ∞. Let (λn ) be a bounded sequence in Rn .
Consider the operator T ∈ L(X , X ) defined by
T x = (λ1 x1 , λ2 x2 , . . . , λn xn , . . .),
where
x = (x1 , x2 , . . . , xn , . . .).
Show that T is a compact operator from X into X if and only if λn → 0.