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A very natural example of a nuclear Montel space is the space $H(D)$ of all holomorphic functions on the open disc topologized by the family of seminorms

$$p_n(f)=\sup\{|f(z)|\colon |z|\leq 1-\tfrac{1}{n}\},\, n\in \mathbb N, f\in H(D) $$

I cannot find any good references concerning this space. In particular, I have two following questions:

1) Can one give examples of subspaces of $H(D)$ which are not isomorphic to it?

2) Does every copy of $H(D)$ contain further complemented one?

Thank you in advance.

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Question 1 is missing the word "infinite-dimensional" –  Yemon Choi Aug 14 '11 at 23:18
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Certainly. Then the next missing word is "closed"... –  RogersFR Aug 14 '11 at 23:38
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Is question one "Does there exists one?" or "How do I write one down?" –  Helge Aug 15 '11 at 19:14
    
Since all the subspaces I can produce are copies of $H(D)$ I ask about the existence of other subspaces but it should be easy (I believe) to construct different ones. –  RogersFR Aug 15 '11 at 19:37
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If $U$ is simply connected and not equal to the whole plane $H(U)$ is isomorphic to $H(D)$ by the Riemann mapping theorem. However, I would conjecture that this is not true if $U$ is simply connected or the whole plane. If $D\subset U$, then $H(U)$ is a subspace of $H(D)$. –  Matthias Ludewig Jun 4 '13 at 7:20

1 Answer 1

There is a simple and natural way to create closed subspaces of $H(D)$ which are not isomorphic to the latter, namely by considering a lacunary sequence $(\lambda_n)$ and letting $E$ be the subspace of functions whose Taylor coefficients vanish away from this sequence. As a concrete example you can take the sequence of squares of the positive integers.

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