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Thread: can such a map be surjective?

  1. #1
    Member Mauritzvdworm's Avatar
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    can such a map be surjective?

    consider the function $\displaystyle f:\mathbb{D}\rightarrow \mathbb{C}$

    where $\displaystyle f$ is an analytic function on the space $\displaystyle \mathbb{D}$ (open unit disk in $\displaystyle \mathbb{C}$)

    can we show that $\displaystyle f$ is in fact surjective?

    In the bigger picture I would like to thow that $\displaystyle \mathcal{A}$, the Banach algebra whose elements are analytic functions on $\displaystyle \mathbb{D}$ with pointwise defined operations is contained in the space $\displaystyle H^{\infty}(\mathbb{D})$ of all bounded analytic function on $\displaystyle \mathbb{D}$

    The plan is as follows:
    Since any $\displaystyle f\in\mathcal{A}$ is analytic it will also be continuous. $\displaystyle \mathbb{D}$ is compact, so if I can show that theses functions $\displaystyle f$ are in fact surjective then I can use preservation of compactness which implies the image $\displaystyle f(\mathbb{D})$ will be compact, and hence closed and bounded.
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  2. #2
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    Quote Originally Posted by Mauritzvdworm View Post

    In the bigger picture I would like to thow that $\displaystyle \mathcal{A}$, the Banach algebra whose elements are analytic functions on $\displaystyle \mathbb{D}$ with pointwise defined operations is contained in the space $\displaystyle H^{\infty}(\mathbb{D})$ of all bounded analytic function on $\displaystyle \mathbb{D}$
    I don't get what you mean in the first paragraph, but this is (I think) false. It is a standard (although not that well known apparently) result that any open set in $\displaystyle \mathbb{C}$ has a function defined on it that cannot be extended at all (Domains of Holomorphy). The proof for this (see Remmert, Classical topics in Complex function theory) builds a function that has a sequence accumulating everywhere in the boundary with $\displaystyle f(x_n) \rightarrow \infty$.

    Also $\displaystyle \mathbb{D}$ is open so it's not compact, which is the problem.

    Edit: Mybe this was overkill (although it proves that this inclusion is not valid for any open set). For your case take $\displaystyle f(z)=\frac{1}{1-z}$ and note that $\displaystyle 1$ is a pole.

    Edit2: Ha, even in the general case it's an overkill. Just take a point in the boundary $\displaystyle p$ and define $\displaystyle f(z)=\frac{1}{p-z}$.
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