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On Sat, 29 Apr 2006 15:44:06 +0100, José Carlos Santos
<jcsantos@xxxxxxxx> wrote:
>On 29-04-2006 15:31, Mate wrote:
>
>>> Let f_n be the linear functionals on l^2
>>>
>>> f_n(x) = ( x_1 + x_2 + ... + x_n ) / n^(1/2).
>>>
>>> It is easy to see that || f_n || = 1.
>>>
>>> Does (f_n) converge poinwise to 0?
>>> Is the problem too easy?
>>
>> Yes, for both questions [I should have think more before posting].
>> (it is sufficient to approximate x with a finite supported sequence).
>
>Why is it sufficient to approximate _x_ with finite supported sequences?
>Yes, it's easy to see that, if _x_ is a finite supported sequence, then
>lim_n f_n(x) = 0, but how do you get the general case from that?
The f_n have bounded norm. Hence they're an equicontinuous family,
(at least when restricted to bounded subsets of l^2) and hence
convergence to 0 on a dense subset implies pointwise convergence to
zero.
>
>Best regards,
>
>Jose Carlos Santos
************************
David C. Ullrich
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