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On Sat, 30 Jul 2005, David C. Ullrich wrote:
> <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_?
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* Generalized series in a normed space by Alex (July 24, 2005)
+ [8]Re: Generalized series in a normed space by mars (July 24,
2005)
o [9]Re: Re: Generalized series in a normed space by Alex
(July 25, 2005)
# [10]Re: Re: Re: Generalized series in a normed
space by mars (July 26, 2005)
@ [11]Re: Re: Re: Re: Generalized series in a
normed space by Alex (July 26, 2005)
- [12]Re: Re: Re: Re: Re: Generalized
series in a normed space by mars (July
26, 2005)
= [13]Re: Re: Re: Re: Re: Re:
Generalized series in a normed space
by Alex (July 26, 2005)
* [14]Re: Re: Re: Re: Re: Re: Re:
Generalized series in a normed space
by mars (July 26, 2005)
_________________________________________________________________
From: Alex
Date: July 24, 2005
Subject: Generalized series in a normed space
Hello,
Before asking my question I'll give the definition of a
convergence of a generalized series in a normed space.
The generalized series Sum_i(x_i), i in some index set I, is
said to converge to x in a normed space X, notation x = Sum_i(x_i),
if for every e > 0 there exist finite subset I_0 of I such that
for every finite superset J of I_0
|| x - Sum_(i in J)(x_i) || < e
Where can I find a proof that this definition is equivalent to:
x = Sum_i(x_i) if and only if
for every e > 0 there exist finite I_0 such that for all finite J
with I_0 /\ J = empty, || Sum_(i in J)(x_i) || < e
First definition is suggestive and intuitively very natural.
I'm trying unsuccessfully to show its equivalence to the second.
Thank you for your help.
Alex
References
1. at.yorku.ca/topology/">http://at.yorku.ca/topology/
2. at.yorku.ca/topology/new.htm">http://at.yorku.ca/topology/new.htm
3. at.yorku.ca/topology/search.htm">http://at.yorku.ca/topology/search.htm
4.
at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3068">http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3068
5.
at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3070">http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3070
6. at.yorku.ca/cgi-bin/bbqa?task=list;forum=ask_an_analyst">http://at.yorku.ca/cgi-bin/bbqa?task=list;forum=ask_an_analyst
7. at.yorku.ca/cgi-bin/bbqa">http://at.yorku.ca/cgi-bin/bbqa
8.
at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3069.0001">http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3069.0001
9.
at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3069.0001.0001">http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3069.0001.0001
10.
at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3069.0001.0001.0001">http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3069.0001.0001.0001
11.
at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3069.0001.0001.0001.0001">http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3069.0001.0001.0001.0001
12.
at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3069.0001.0001.0001.0001.0001">http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3069.0001.0001.0001.0001.0001
13.
at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3069.0001.0001.0001.0001.0001.0001">http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3069.0001.0001.0001.0001.0001.0001
14.
at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3069.0001.0001.0001.0001.0001.0001.0001">http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=3069.0001.0001.0001.0001.0001.0001.0001
15. at.yorku.ca/topology/">http://at.yorku.ca/topology/
16. users.binary.net/dturley/">http://users.binary.net/dturley/
--
> 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?
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