Abstract
We want to prove, as in the case of the disk, that H(G)* = H 0(ℂ \ G). We first study the dual of C(G). We change our notation here and write L(f) = ∫ fdμ when L ∈ C(G)*. (For the reader unfamiliar with integration theory this is simply a change in notation: The left-hand side defines the right-hand side. There are two advantages to this notation. First, it is the notation in which research papers are written. Second, the reader can call upon her experience with integration for intuition. For the mathematically advanced reader: we are invoking the Riesz Representation Theorem for C(G)*.) We call μ the “measure” associated with L, and we may identify μ and L. The collection of all such μ is denoted M0(G), so that M0(G) = C(G)*. We also write L(f) = ∫ f(z)dμ(z) when it is necessary to indicate the independent variable. “Measures” have the same properties as continuous linear functionals (which is what they are); for reinforcement, we list them here. Given μ ∈ M0(G):
-
i)
∫ (f + g)dμ = ∫ fdμ + ∫ gdμ, f, g ∈ C(G).
-
ii)
∫ afdμ = a ∫ fdμ, f ∈ C(G), a ∈ ℂ.
-
iii)
If fn → f in C(G) then ∫ fndμ → ∫ fdμ.
-
iv)
There is a compact set K ⊆ G such that | ∫ fdμ | ≤ C‖f‖K for all f ∈ C(G).
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer-Verlag New York Inc.
About this chapter
Cite this chapter
Luecking, D.H., Rubel, L.A. (1984). The Dual of H(G). In: Complex Analysis. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8295-9_9
Download citation
DOI: https://doi.org/10.1007/978-1-4613-8295-9_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-90993-6
Online ISBN: 978-1-4613-8295-9
eBook Packages: Springer Book Archive