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Let $H(\mathbb{C})=\{f: \mathbb{C}\rightarrow \mathbb{C}; f \text{ holomorphic}\}$. For each $n$ let the seminorm $p_n$ be $p_n(f)=\sup_{|z|\leq n}|f(z)|$, and let $d(f,g)=\sum_{n=1}^\infty \frac{1}{2^n} \min (1,p_n(f-g))$. I am to show that $H(\mathbb{C})$ is a Frechet space under the metric $d$.

My only issue is showing $H(\mathbb{C})$ is complete under $d$. Given a Cauchy sequence $(f_k)$, I can show that the pointwise limit exists, but I am having trouble showing this limit is in $H(\mathbb{C})$. Any advice?

Thank you.

user122916
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  • Answer can be found in http://math.stackexchange.com/questions/368664/uniform-limit-of-holomorphic-functions –  Nov 22 '14 at 01:59

1 Answers1

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If a sequence $(f_k)$ in $H(\mathbb{C})$ is Cauchy with respect to the given family of seminorms, then $(f_k)$ converges uniformly on each closed ball $B(0,n)$. As a consequence of Morera's theorem, the limit function $f$ is holomorphic on each closed ball $B(0,n)$.

Gyu Eun Lee
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  • Can you expand a bit more on your first sentence? Thank you. – user122916 Nov 22 '14 at 02:41
  • The definition of being $p_n$-Cauchy is that the uniform norm $\sup_{|z|\leq n}|f_m(z)-f_n(z)|$ goes to $0$ as $m,n\to\infty$, which is to say $(f_m)$ is a uniformly Cauchy sequence in $B(0,n)$. One can show that $(f_m)$ is Cauchy in the $d$-"metric" if and only if it is Cauchy in each seminorm $p_n$. (The $\min(1,p_n)$ is only there to ensure that $d$ is well-defined; for the purposes of determining the Cauchy criterion the $1$ is rather irrelevant.) – Gyu Eun Lee Nov 22 '14 at 06:04
  • And do the $(f_m)$ converge uniformly to something because $B(0,n)$ is compact? – user122916 Nov 24 '14 at 09:04
  • No, they converge uniformly by definition of what it means to converge in the seminorm $p_n$. – Gyu Eun Lee Nov 24 '14 at 09:13
  • But we don't know that they converge; just that they are Cauchy. Could you show explicitly that they converge? – user122916 Dec 03 '14 at 16:28
  • Clearly they converge to a continuous function, and you can use Morera's theorem to show the limit is holomorphic; see John's comment. So you really are done once you establish that they are Cauchy. – Gyu Eun Lee Dec 04 '14 at 19:02