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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."
Let's make things clear and give the definition of a convergence of a
generalized series.
In analogy with number series which are special
kind of sequences(sequences of partial sums), we can proceed in the same way
with nets and define
generalized series as a net of partial sums.
This is still for the moment not very clear.
Let (x_i), i in I, be a net in a topological space X.
A := {F |F subset I, F is finite }.
(A,<=) is directed with
F_1 <= F_2 if F1 is a subset of F_2.
We consider the net in X
A --> X, F |--> Sum(i in F)x_i =: S_F
This net (S_F) in X is called generalized series in X,
and is usually denoted by Sum(i in I)x_i.
By definition Sum(i in I)x_i converges if the net
(S_F) converges. Net convergence is well-known.
> 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.)
Are we talking now about the same kind of convergence?
David, can you please give the proof of this?
Now, necessary condition for the convergence
of Sum(i in I)x_i is (x_i) to be zero-net.
But I still doubt that another necessary condition
will be x_i =/= 0 for at most countably many i's.
Xantos
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