Measure, Integration & Real Analysis, 2021a

254 Chapter 8 Hilbert Spaces
23 Prove that if f ∈ L 2 a(D \{0}), then f has a removable singularity at 0 (meaning
that f can be extended to a function that is analytic on D).
24 The Dirichlet space D is defined to be the set of analytic functions f : D → C
such that
∫
| f ′ | 2 dλ 2 < ∞.
For f , g ∈D, define 〈 f , g〉 to be f (0)g(0)+ ∫ D f ′ g ′ dλ 2 .
D
(a) Show that D is a Hilbert space.
(b) Show that if w ∈ D, then f ↦→ f (w) is a bounded linear functional on D.
(c) Find an orthonormal basis of D.
(d) Suppose f ∈Dhas Taylor series
f (z) =
∞
∑ a k z k
k=0
for z ∈ D. Find a formula for ‖ f ‖ in terms of a 0 , a 1 , a 2 ,....
(e) Suppose w ∈ D. Find an explicit formula for Γ w ∈Dsuch that
f (w) =〈 f , Γ w 〉 for all f ∈D.
25 (a) Prove that the Dirichlet space D is contained in the Bergman space L 2 a(D).
(b) Prove that there exists a function f ∈ L 2 a(D) such that f is uniformly
continuous on D and f /∈ D.
Measure, Integration & Real Analysis, by Sheldon Axler