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On Sat, 30 Jul 2005 06:45:32 -0700, William Elliot
<marsh@xxxxxxxxxxxxxxxxxx> wrote:
>On Sat, 30 Jul 2005, David C. Ullrich wrote:
>> On Sat, 30 Jul 2005 07:42:54 EDT, Xantos <xantos_2005@xxxxxxxxx>
>> wrote:
>>
>> >Nets are generalized sequences. In the same manner
>> >series can be generalized. I don't know it they also
>> >have some fancy name.
>> >
>> >If the generalized series Sum_i(x_i) converges,
>>
>> It's not at all clear to me what you mean by this,
>> since (as we've seen elsewhere in sci.math recently)
>> the "partial sums" here can be infinite sums.
>>
>It's not clear to me either. This generalized series stuff was defined
>with clearer stated problem and my (mars) belabored solutions at
>http://at.yorku.ca/topology in the Ask-an-Analysis form within the thread
>"Generalized series in normed vector spaces."
I don't see a link to such a forum, nor to a group of
forums there - searching for "Generalized series in normed
vector spaces" gives no hits. Why don't you find the
article you're referring to and give a specific link
to _it_?
Meanwhile, what I _suspect_ you're referring to is
a definition of sum_{i in A} x_i where A is a _set_.
There's a fairly standard definition of that
(and in that context it's easy to show that
convergence of a sum implies that only countably
many terms are non-zero.)
If that's what you're referring to it can't be
exactly what we're talking about here, because
it talks about sums of things indexed by elements
of a _set_, not a net.
>> >then it is easy to show that the net
>> >(x_i) is a zero-net.
>> >
>> >But what I really would like to know is whether in that case at most
>> >countably many x_i's are different from zero. That is to say, the
>> >summation can be done only on countable subset of I.
>
>That is correct for R^n and I found no counter examples for R^N and R^R.
What's the _definition_ of Sum_i(x_i) you have in mind here?
>See reference above for proofs, discussion and clear and precise
>definitions and problem statement
************************
David C. Ullrich
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