Functional Analysis

Math 511 Problem Set 6, due October 12

1. We say that a linear operator T : X → Y between two normed spaces is bounded below if there is some b > 0 so that for every x ∈ X, ‖Tx‖ ≥ b‖x‖. Suppose that T ∈ B(X,Y ) is bounded below by b > 0.

(a) Show that T is injective.

(b) Show that if X is a Banach space, then range(T) is a closed subspace of Y .

(c) Show that T−1 : range(T) → X is a bounded linear operator and ‖T−1‖≤ 1 b .

2. Let X be a vector space over field F and Y a subspace. If x ∈ X, the coset of Y containing x is the set x + Y = {x + y : y ∈ Y}. Denote the set of all cosets of Y to be X/Y . Define an addition and scalar multiplication on X/Y by if x1 + Y,x2 + Y ∈ X/Y and k ∈ F, then

(x1 + Y ) + (x2 + Y ) = (x1 + x2) + Y and k(x1 + Y ) = (kx1) + Y.

(a) Show that these operations are well-defined, and these make X/Y into a vector space.

(b) Define the codimension of Y to be codim(Y ) = dim(X/Y ). Prove that if f is a linear functional on X, then codim(ker(f)) = 1.

3. Suppose that X is a linear space and Y is a subspace. You showed above that X/Y is a vector space. Now suppose additionally that X has a norm and Y is a closed subspace. Show that ‖x + Y‖ = inf{‖x + y‖ : y ∈ Y} defines a norm on X/Y . Note: It also follows that if X is a Banach space then X/Y is also a Banach space. The proof can be found online.

4. Let X = C[0, 1] with the supremum norm, ‖ · ‖∞. Consider the functional φ defined for f ∈ X by

φ(f) =

∫ 1 2

0

f(x) dx− ∫ 1

1 2

f(x) dx.

(a) Prove that φ is a bounded linear functional with norm 1.

(b) Show that if f ∈ X with ‖f‖ = 1 then |φ(f)| < 1. (c) Briefly describe how this relates to Riesz’s lemma (2.5-4). (Hint: φ is continuous, so

ker(φ) is a closed subspace of X.)