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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Summer 2021 • Question 2

Question

Math 6220 • Complex Analysis Summer 2021 • Question 2

Let \(U\) be a domain in \(\mathbb {C}\). Consider the following collection of functions \[ X=\{f:U\to \mathbb {C}: f\text { holomorphic and }\int _U |f|^2\le M\} \]

where \(\int _U\) is the standard Lebesgue area integral. Show that \(X\) is compact with respect to the topology of uniform convergence on compact subsets of \(U\) (Note: this topology is metrizable so you can show sequential compact- ness).

Raw LaTeX

Let \(U\) be a domain in \(\mathbb{C}\). Consider the following collection of functions
\[
X=\{f:U\to\mathbb{C}: f\text{ holomorphic and }\int_U |f|^2\le M\}
\]

where \(\int_U\) is the standard Lebesgue area integral. Show that \(X\) is compact
with respect to the topology of uniform convergence on compact subsets of
\(U\) (Note: this topology is metrizable so you can show sequential compact-
ness).

Topic Tags

Cauchy Integral Theory and Morera's Theorem Complex Integration and Real Integral Evaluation Holomorphic Functions and Cauchy-Riemann Equations Uniform Convergence and Hurwitz's Theorem