I'm working in the metric space $l^{\infty}$ trying to show that a certain subset is not complete. So I construct a sequence of sequences $(x_n)=(x_1,x_2,\ldots)$ where for each $x_n \in (x_n)$ $$x_n=(1,1/2,1/3,\ldots,1/n,0,0,0,0,0,\ldots)$$ The confusion creeps in when I look at an example in Kreyszig-"Introductory Functional Analysis with Applications" where he defines an arbitrary sequence of sequences as$$(x_m)=(\zeta_1^{(m)},\zeta_{2}^{(m)},\zeta_3^{(m)},\ldots)$$ As he wants to prove that it's Cauchy he goes on to say that $$d(x_m,x_n)=\sup_{j \in \mathbb{N}}|x_j^{(n)}-x_j^{(m)}|$$ Now I do not understand where he's going with this notation, as it seems that he's defining the $m$'th element of his sequence as the sequence taking the $m$'th element of $\zeta_1$ as the first element, and the $m$'th element of $\zeta_2$ as the 2nd element and so on and so forth.

This seems to me to be a rather unnatural way of going about it, as I'd like to just define the sequence such that the $m$'th element is simply $\zeta_m$. In fact that's how I need to define my original sequence to get the result I need.

So my question is: Is my simplistic way of defining a sequence in a sequence space correct? If so how should I notate it so I can express the metric properly when dealing with $x_m$ and $x_n$? Or am I just completely misunderstanding Kreyszig's notation and he is expressing what I want to?

Specifically for my sequence I want $(1,0,0,0,\ldots)$ to be the first element of the sequence, $(1,1/2,0,0,0,\ldots)$ to be the second element and $(1,1/2,1/3,\ldots,1/n,0,0,0,\ldots$ to be the n'th element of the sequence, and so on. What notation should I use to describe this sequence?


1 Answer 1


I believe you are misunderstanding Kreyzig's notation. When he uses the index m, he means the m'th sequence of the sequence of sequences.

... as the sequence taking the $m$'th element of $\zeta_1$ as the first element

Here is the root of your misunderstanding, $\zeta_1$ is nothing here (unless you define it), $\zeta_1^{(m)}$ is the first element of the $m$'th sequence.

I'm not sure if you understood the wanted metric, he is comparing for these two sequences $x_m$ and $x_n$, the difference between each coordinate.

  • $\begingroup$ Ok I'll edit my question to define what I want from my sequence and if you can show me how to notate it I'd be much obliged. $\endgroup$
    – K.Power
    Feb 23, 2016 at 17:43
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    $\begingroup$ Oh you want only a good looking way to write it? I mean, you can always write in words (as you described), I believe there is no need to use too much mathematical notations if in words is simpler. Another way I see it could be is to define the canonical vector $(e_m)= (0,0,...,1,0,0,.. )$ and then your sequences would be $(x_m)= \sum_{n=1}^m e_n \frac{1}{n}$. And this should be super friendly for checking non-Cauchyness $\endgroup$
    – pancho
    Feb 23, 2016 at 18:06
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    $\begingroup$ Thanks so much, that's what I was looking for! I do understand Kreyszig's notation now too so thank you. $\endgroup$
    – K.Power
    Feb 23, 2016 at 19:17

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