Functional Analysis

Math 511 Problem Set 5, due October 5

1. Let X be a normed linear space and Y a finite dimensional subspace of X so that Y 6= X. Prove that there is a z ∈ X\Y with ‖z‖ = 1 such that for any y ∈ Y , ‖z − y‖ ≥ 1. (Hint: Pick some z0 ∈ X\Y , and let Z = span{Y, z0}. Since Y is finite dimensional, then Z is finite dimensional, and use Riesz’s Lemma and the compactness of the closed unit ball of Z.)

2. Suppose that X and Y are vector spaces and T : X → Y is an invertible linear operator. Show that if {x1, x2, . . . , xn} is a linearly independent set in X, then {Tx1, Tx2, . . . , Txn} is linearly independent in Y .

3. Define the linear map T : l∞ → l∞ by Tx = ( xk k

) , where x = (xk). Show that T is a bounded

linear operator and is injective. Is T−1 bounded? (Note: The domain of T−1 may not be all of l∞)

4. Let X, Y , and Z be normed linear spaces. For S ∈ B(X, Y ) and T ∈ B(Y, Z) define the linear operator TS : X → Z by function composition: TSx = T ◦ S(x). Prove that ‖TS‖≤‖T‖‖S‖.

5. Let X = R2 with the norm ‖ · ‖1 so that ‖(y, z)‖1 = |y| + |z|. Fix some a ∈ R, and define T ∈ B(X, X) by

T (y, z) =

[ 1 a 0 1

] [ y z

] .

Prove that ‖T‖ = 1 + |a|. Observe the difference in this case with the example from class in that the norm of an operator depends on the norms of its domain and codomain.

6. Consider the operator A : l2 → l2 given by x = (xn) ∈ l2 implies Ax = ( xnn ). Show that range(A) is a dense subspace of l2 but not closed.