Measure, Integration & Real Analysis · Problem 8C.20

Real and Complex Measures — Sheldon Axler

Mathematics — Measure Theory · bergman-space

Problem

Suppose GG is a nonempty open subset of C\mathbf{C}. The Bergman space La2(G)L^2_a(G) is defined to be the set of analytic functions f:G→Cf: G \to \mathbf{C} such that ∫G∣f∣2 dλ2<∞,\int_G |f|^2\, d\lambda_2 < \infty, where λ2\lambda_2 is the usual Lebesgue measure on R2\mathbf{R}^2, which is identified with C\mathbf{C}. For f,h∈La2(G)f, h \in L^2_a(G), define ⟨f,h⟩\langle f, h\rangle to be ∫Gfhˉ dλ2\int_G f\bar{h}\, d\lambda_2. (a) Show that La2(G)L^2_a(G) is a Hilbert space. (b) Show that if w∈Gw \in G, then f↦f(w)f \mapsto f(w) is a bounded linear functional on La2(G)L^2_a(G).

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