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Banach limit

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Mathematics Subject Classification

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I have to questions about Banach limits. Maybe someone of you knows something about it or could provide at least a pointer, where I could find information I'm interested in.

1. I'm aware of two possible proofs of existence of Banach limits. One of them employs ultrafilters, the other one uses Hahn-Banach theorem. How much Choice is really needed for existence of Ban. limit? (Maybe this example shows better what I want to know: It's known that equivalence of Heine's and Cauchy's definition of continuity implies Countable Choice. Is some similar result known for Banach limits?)

2. Was analogous concept defined also for some more general setting, e.g. in Banach spaces?


For question #2:

Yes, although I am not too familiar with it.
There is a notion of a "generalized limit"
in Yoshida's Functional Analysis book, which uses
nets rather than l^\infty sequences.
So I would think that the procedure
employed, for example, to extend the
Riemann integral to the Lebesgue integral.

(note Riemann integral is a limit I
st. L <= I <= U for the supremum L of
the lower sums and the infimum U of the upper sums.
The situation seems to be analogous to the liminf
and limsup of a sequence.)

If you (or anyone else) would like to elaborate
on the current article, I can give you write access.

// Steve

I would say that nets is another type of generalized limit. What I mean is this: nets are like sequences, but instead of N they are indexed by an upwards directed set. It means, we changed the "domain" of the sequence in this generalization - the notion of sequence is replaced with the notion of net in this generalization of limit. Nets can be defined for any topological space.

In the case of Banach limit, the notion of sequence isn't changed, but this operator assigns the value also to some non-convergent sequences.

> If you (or anyone else) would like to elaborate
> on the current article, I can give you write access.
There already is entry on nets in topological spaces - (Although it doesn't mention the example with the Riemann integral.)


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