Name that Theorem!

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Name that Theorem! James Currie 12/6/95 12:00 AM
Could someone please give references for the following result? What
extensions etc. are known?

Theorem: Let f(x) be a real valued function of one real variable such
that \sum_{n=1}^\infty f(a_n) converges whenever \sum_{n=1}^\infty a_n
converges. Then f(x) = cx, some constant c, on some neighbourhood of x =
0.

James Currie, Math
cur...@uwinnipeg.ca


Name that Theorem! Gareth McCaughan 12/6/95 12:00 AM
In article <Pine.OSF.3.91.951206111141.1482A-100000@io.UWinnipeg.ca>,
James Currie  <cur...@io.UWinnipeg.ca> wrote:

>Theorem: Let f(x) be a real valued function of one real variable such
>that \sum_{n=1}^\infty f(a_n) converges whenever \sum_{n=1}^\infty a_n
>converges. Then f(x) = cx, some constant c, on some neighbourhood of x =
>0.

Presumably "whenever" means "if and only if" here? If f(x) is always 0
then sum f(a_n) certainly converges whenever sum a_n converges...

--
Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
gj...@pmms.cam.ac.uk Cambridge University, England.    [Research student]


Name that Theorem! Robert Israel 12/6/95 12:00 AM
In article <Pine.OSF.3.91.951206111141.1482A-100000@io.UWinnipeg.ca>, James Currie <cur...@io.UWinnipeg.ca> writes:
|> Could someone please give references for the following result? What
|> extensions etc. are known?
|>
|> Theorem: Let f(x) be a real valued function of one real variable such
|> that \sum_{n=1}^\infty f(a_n) converges whenever \sum_{n=1}^\infty a_n
|> converges. Then f(x) = cx, some constant c, on some neighbourhood of x =
|> 0.

This was proved by G. Waldenberg, American Mathematical Monthly 95 (1988)
542-544. Y. Benjamini's solution to Problem E3404, American Mathematical
Monthly 99 (1992) 466-467 contains an extension:

Let f: X -> Y be a mapping of normed spaces such that \sum_{n=1}^\infty f(a_n) converges whenever \sum_{n=1}^\infty a_n converges (both in the norm topology).
Then there is a neighbourhood of 0 on which f is equal to a bounded linear
operator.

I have a slight additional extension (unpublished):

Let f: X -> Y be a mapping of Banach spaces such that \sum_{n=1}^\infty f(a_n) converges weakly whenever \sum_{n=1}^\infty a_n converges (strongly).  Then
there is a neighbourhood of 0 on which f is equal to a bounded linear
operator.

--
Robert Israel                            isr...@math.ubc.ca
Department of Mathematics             (604) 822-3629
University of British Columbia            fax 822-6074
Vancouver, BC, Canada V6T 1Y4


Name that Theorem! Gareth McCaughan 12/7/95 12:00 AM
In article <4a4nb3$3...@lyra.csx.cam.ac.uk>, I wrote:

> In article <Pine.OSF.3.91.951206111141.1482A-100000@io.UWinnipeg.ca>,


> James Currie  <cur...@io.UWinnipeg.ca> wrote:
>
> >Theorem: Let f(x) be a real valued function of one real variable such
> >that \sum_{n=1}^\infty f(a_n) converges whenever \sum_{n=1}^\infty a_n
> >converges. Then f(x) = cx, some constant c, on some neighbourhood of x =
> >0.
>
> Presumably "whenever" means "if and only if" here? If f(x) is always 0
> then sum f(a_n) certainly converges whenever sum a_n converges...

A couple of people mailed me to point out that this is just fine, with c=0.
My thought processes had gone something like "well, you can take f(x)=o(x)
and surely that will do. What's the simplest case? Ah, f=0."

However, I must apologise very completely, because even this is not
true (so my previous posting was wholly without foundation). For instance,
f(x)=x^3 won't do, because you could (for instance) let b_n=n^{-1/4}
and then have a_n go b1,-b1/2,-b1/2,b2,-b2/2,-b2/2,b3,-b3/2,-b3/2,...
whose sum obviously converges to 0, while the sum of f(a_n) is
(1-2^{3/4})(b1^3+b2^3+b3^3+...) and this obviously diverges.

So, I grovel. One of these years I'll manage to post something to
sci.math.research without silly mistakes in. Maybe.

--
Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
gj...@pmms.cam.ac.uk Cambridge University, England.    [Research student]