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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2002 • Question 9

Question

Math 6210 • Real Analysis 2002 • Question 9

Let \(D\) be a domain in the plane, and \(\{f_n\}\) a sequence of functions holomorphic on \(D\). Suppose the sequence is uniformly bounded; that is, there is a positive number \(M\) such that \[ |f_n(z)|<M \quad \text {for all } n \text { and } z\in D. \] Suppose, in addition, that there is an infinite compact set \(K\) in \(D\) such that the sequence \(f_n|K\) converges uniformly on \(K\). Show that the sequence \(f_n\) converges uniformly on all compact subsets of \(D\).

Raw LaTeX

Let \(D\) be a domain in the plane, and \(\{f_n\}\) a sequence of
functions holomorphic on \(D\). Suppose the sequence is uniformly bounded; that
is, there is a positive number \(M\) such that
\[
|f_n(z)|<M \quad\text{for all } n \text{ and } z\in D.
\]
Suppose, in addition, that there is an infinite compact set \(K\) in \(D\) such that
the sequence \(f_n|K\) converges uniformly on \(K\). Show that the sequence \(f_n\)
converges uniformly on all compact subsets of \(D\).

Topic Tags

Compactness in Metric and Normed Spaces Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Lp Spaces and Sequence Spaces