The Dual of H(G)

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 f_n → f in C(G) then ∫ f_ndμ → ∫ fdμ.

• iv) There is a compact set K ⊆ G such that | ∫ fdμ | ≤ C‖f‖_K for all f ∈ C(G).