## 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? Whatextensions 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, Mathcur...@uwinnipeg.ca Name that Theorem! Gareth McCaughan 12/6/95 12:00 AM In article ,James Currie   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 0then 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 , James Currie 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 MathematicalMonthly 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there is a neighbourhood of 0 on which f is equal to a bounded linear operator.I have a slight additional extension (unpublished): Let f: X -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).  Thenthere is a neighbourhood of 0 on which f is equal to a bounded linear operator.-- Robert Israel                            isr...@math.ubc.caDepartment of Mathematics             (604) 822-3629University of British Columbia            fax 822-6074Vancouver, 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 ,> James Currie   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 nottrue (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 tosci.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]