9. Dual Spaces

Question 1

Define what is meant by a bounded linear functional on X and define the dual space X∗ of X.

Answer 1

Definition 9.1: A bounded linear functional on X is a map φ : X → F which is linear (i.e. φ(λx + μy) = λφ(x) + μφ(y) for all x, y ∈ X and all λ, μ ∈ F) and bounded (i.e. there exists a constant M ≥ 0 such that |φ(x)| ≤ M∥x∥ for all x ∈ X). The dual space X∗ is the set of all bounded linear functionals on X.

Question 2

Define the operations of addition and scalar multiplication on
∗
X

Answer 2

(φ + ψ)(x) = φ(x) + ψ(x)
for φ,ψ ∈ X∗, x ∈ X, and
(λφ)(x) = λφ(x)
for all φ ∈ X∗, λ ∈ F, x ∈ X.

Question 3

Define the usual norm on X .

Answer 3

The norm of φ ∈ X∗ is defined as
∥φ∥=sup{|φ(x)|:x∈X, ∥x∥≤1}.

Question 4

State the Riesz representation theorem

Answer 4

Let X be a Hilbert space. For every y ∈ X the map x 􏰀→ ⟨x, y⟩ defines a bounded linear functional of norm ∥y∥ on X. Conversely, for every φ ∈ X∗ there exists a unique y ∈ X such that
φ(x) = ⟨x,y⟩, x ∈ X.