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Let $H(D)$ be the set of all analytic functions in a region $D$ in $C$ or in $C^n$. Everyone who worked with this set knows that there is only one reasonable topology on it: the uniform convergence on compact subsets of $D$.

Is this statement correct?

If yes, can one state and prove a theorem of this sort:

If a topology on $H(D)$ has such and such natural properties, then it must be the topology of uniform convergence on compact subsets.

One natural property which immediately comes in mind is that the point evaluatons must be continuous. What else?

EDIT: 1. On my first question, I want to add that other topologies were also studied. For example, pointwise convergence (Montel, Keldysh and others). Still we probably all feel that the standard topology is the most natural one.

2. If the topology is assumed to come from some metric, a natural assumption would be that the space is complete. But I would not like to assume that this is a metric space a priori.

3. As other desirable properties, the candidates are continuity of addition and multiplication. However it is better not to take these as axioms, because the topology of uniform convergence on compacts is also the most natural one for the space of meromorphic functions (of one variable), I mean uniform with respect to the spherical metric.

4. Whenever I try to enumerate some paragraphs with 1, 2, 3..., the MathJack produces some nonsense as you can see in this message.

Let $H(D)$ be the set of all analytic functions in a region $D$ in $C$ or in $C^n$. Everyone who worked with this set knows that there is only one reasonable topology on it: the uniform convergence on compact subsets of $D$.

Is this statement correct?

If yes, can one state and prove a theorem of this sort:

If a topology on $H(D)$ has such and such natural properties, then it must be the topology of uniform convergence on compact subsets.

One natural property which immediately comes in mind is that the point evaluatons must be continuous. What else?

EDIT: 1. On my first question, I want to add that other topologies were also studied. For example, pointwise convergence (Montel, Keldysh and others). Still we probably all feel that the standard topology is the most natural one.

2. If the topology is assumed to come from some metric, a natural assumption would be that the space is complete. But I would not like to assume that this is a metric space a priori.

3. As other desirable properties, the candidates are continuity of addition and multiplication. However it is better not to take these as axioms, because the topology of uniform convergence on compacts is also the most natural one for the space of meromorphic functions (of one variable), I mean uniform with respect to the spherical metric.

Let $H(D)$ be the set of all analytic functions in a region $D$ in $C$ or in $C^n$. Everyone who worked with this set knows that there is only one reasonable topology on it: the uniform convergence on compact subsets of $D$.

Is this statement correct?

If yes, can one state and prove a theorem of this sort:

If a topology on $H(D)$ has such and such natural properties, then it must be the topology of uniform convergence on compact subsets.

One natural property which immediately comes in mind is that the point evaluatons must be continuous. What else?

EDIT: 1. On my first question, I want to add that other topologies were also studied. For example, pointwise convergence (Montel, Keldysh and others). Still we probably all feel that the standard topology is the most natural one.

2. If point evaluations are continuous, convergence in such a topology implies pointwise convergence. However, pointwise limit of analytic functions is not always analytic, and one may want to state a desirable property that a limit in our topology of analytic functions is analytic. Will this be enough for characterization of the standard topology?

3. As other desirable properties, the candidates are continuity of addition and multiplication. However it is better not to take these as axioms, because the topology of uniform convergence on compacts is also the most natural one for the space of meromorphic functions (of one variable), I mean uniform with respect to the spherical metric.

Let $H(D)$ be the set of all analytic functions in a region $D$ in $C$ or in $C^n$. Everyone who worked with this set knows that there is only one reasonable topology on it: the uniform convergence on compact subsets of $D$.

Is this statement correct?

If yes, can one state and prove a theorem of this sort:

If a topology on $H(D)$ has such and such natural properties, then it must be the topology of uniform convergence on compact subsets.

One natural property which immediately comes in mind is that the point evaluatons must be continuous. What else?

EDIT: 1. On my first question, I want to add that other topologies were also studied. For example, pointwise convergence (Montel, Keldysh and others). Still we probably all feel that the standard topology is the most natural one.

2. If the topology is assumed to come from some metric, a natural assumption would be that the space is complete. But I would not like to assume that this is a metric space a priori.

3. As other desirable properties, the candidates are continuity of addition and multiplication. However it is better not to take these as axioms, because the topology of uniform convergence on compacts is also the most natural one for the space of meromorphic functions (of one variable), I mean uniform with respect to the spherical metric.

4. Whenever I try to enumerate some paragraphs with 1, 2, 3..., the MathJack produces some nonsense as you can see in this message.

Let $H(D)$ be the set of all analytic functions in a region $D$ in $C$ or in $C^n$. Everyone who worked with this set knows that there is only one reasonable topology on it: the uniform convergence on compact subsets of $D$.

Is this statement correct?

If yes, can one state and prove a theorem of this sort:

If a topology on $H(D)$ has such and such natural properties, then it must be the topology of uniform convergence on compact subsets.

One natural property which immediately comes in mind is that the point evaluatons must be continuous. What else?

Let $H(D)$ be the set of all analytic functions in a region $D$ in $C$ or in $C^n$. Everyone who worked with this set knows that there is only one reasonable topology on it: the uniform convergence on compact subsets of $D$.

Is this statement correct?

If yes, can one state and prove a theorem of this sort:

If a topology on $H(D)$ has such and such natural properties, then it must be the topology of uniform convergence on compact subsets.

One natural property which immediately comes in mind is that the point evaluatons must be continuous. What else?

EDIT: 1. On my first question, I want to add that other topologies were also studied. For example, pointwise convergence (Montel, Keldysh and others). Still we probably all feel that the standard topology is the most natural one.

2. If point evaluations are continuous, convergence in such a topology implies pointwise convergence. However, pointwise limit of analytic functions is not always analytic, and one may want to state a desirable property that a limit in our topology of analytic functions is analytic. Will this be enough for characterization of the standard topology?

3. As other desirable properties, the candidates are continuity of addition and multiplication. However it is better not to take these as axioms, because the topology of uniform convergence on compacts is also the most natural one for the space of meromorphic functions (of one variable), I mean uniform with respect to the spherical metric.