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August 14[edit]

Convergence Question[edit]

Consider the following complex-valued two-way sequences: with the property that converges. Suppose you have a sequence of such sequences which and converges to Show that converges too. Widener (talk) 12:34, 14 August 2012 (UTC)[reply]

How are you defining convergence in the space of sequences? Rckrone (talk) 21:48, 14 August 2012 (UTC)[reply]
They're convergent sequences, so presumably you just identify them with their limits. Any other definition would seen strange to me. --Tango (talk) 22:49, 14 August 2012 (UTC)[reply]
I think it means pointwise convergence. So if , then

. Of course, in this interpretation, the stated result is false. Define if and otherwise.--121.73.35.181

Sorry. It does not mean pointwise convergence; it means convergence under the metric . --Widener (talk) 00:06, 15 August 2012 (UTC)(talk) 23:54, 14 August 2012 (UTC)[reply]
You should think of using the triangle inequality with . Sławomir Biały (talk) 01:40, 15 August 2012 (UTC)[reply]
If it helps: a sequence such that converges must converge to ____? Sławomir Biały (talk) 22:41, 14 August 2012 (UTC)[reply]
I misunderstood your question. This observation will not help. Sławomir Biały (talk) 01:37, 15 August 2012 (UTC)[reply]


So you are dealing with a convergent sequence in the Hilbert space . A convergent sequence is bounded, and the norm is continuous. So (which you can prove directly by the triangle inequality as suggested above). --pma 09:39, 15 August 2012 (UTC)[reply]
Thanks ! 00:28, 16 August 2012 (UTC)Widener (talk)

Example of Limitation of Riemann integration[edit]

How do You show That is Not a Fourier series Of a Riemann integrable Function? Widener (talk) 14:36, 14 August 2012 (UTC)[reply]

Why don't you start by trying to explain why
is a Fourier series of a Riemann integrable function? Fly by Night (talk) 17:17, 14 August 2012 (UTC)[reply]
Is it because the sequence of Fourier coefficients is not in ? Widener (talk) 23:06, 14 August 2012 (UTC)[reply]
Yes. Sławomir Biały (talk) 00:19, 15 August 2012 (UTC)[reply]