Talk:Fréchet Space (Functional Analysis) is Complete Metric Space

Why is it not clear that $\sequence {x^n}_{n \mathop \in \N}$ is a sequence in $\R^\omega$?
Hi prime mover, thank you for reviewing. What is still not clear and what would be your suggestion?

It was already the best clearly stated that
 * Let $\sequence {x^n}_{n \mathop \in \N} := \tuple {x^0, x^1, x^2, \ldots}$ be a Cauchy sequence in $\R^\omega$.


 * So why use $x^n$ for it?
 * Because $x_n$ is already used. What alternative do you recommend?
 * Used for what? I don't see $x_n$ used anywhere.
 * Please check the definition of $x\in\R^\omega$ in Definition:Fréchet Space (Functional Analysis) $x_n$ is used there. --Usagiop
 * You could always try explaining what it means. And saying $x^n$ means $\tuple {x^0, x^1, x^2 \ldots}$ doesn't help much I'm afraid. --prime mover (talk) 10:46, 27 May 2022 (UTC)

A Cauchy sequence is just a Cauchy sequence, not more or less. $x^n$ denotes the $n$-the sequence element, not a power of some undefined symbol $x$. In the space $\R ^\omega$, there is no multiplication, no power.

The expression $\sequence {x^n}_{n \mathop \in \N}$ has the same meaning as $\sequence {x_n}_{n \mathop \in \N}$. I am using the former instead of the latter, only since the latter is already used to express the elements in $\R ^\omega$.


 * That's where the problem is. Using $x^n$ when you really mean $x_n$ is ridiculous.
 * I agree that super scripts are confusing for some people but it is a common idea. In the field where you need nested indexing like in the differential geometry or some mathematical physics, one needs both super- and subscripts for indexing.

Now, I improved a Cauchy sequence to an arbitrary Cauchy sequence.

If this is still not satisfactory, please give me a suggestion for better notation.


 * Don't use $x^n$ for something that does not mean $x^n$, or if you do use it, don't explain it in terms of $x$ with the subscript of a sequence of numbers.
 * If I may not use $x^n$, what should I use instead? Please give me a concrete proposal.
 * Something that can be explained. If you can't explain what the notation means, then there are two possibilities. One is that the notation is bad. --prime mover (talk) 10:46, 27 May 2022 (UTC)
 * I explained that it is an arbitrary Cauchy sequence. --Usagiop

How about $\sequence {x^{(n)} }_{n \mathop \in \N}$ for a sequence in $\R^\omega$?

Then each element $x^{\paren n}$ expands
 * $\sequence { {x^{\paren n}} _i}_{i \mathop \in \N} $

---Usagiop


 * Why not use the notation of whatever source work you got the proof from? --prime mover (talk) 10:47, 27 May 2022 (UTC)


 * $x^{\paren n}$ is standard notation for a sequence of sequences, btw Caliburn (talk) 13:51, 27 May 2022 (UTC)


 * Could you please give me first your concrete proposal? It really helps. --Usagiop


 * I don't have a concrete proposal. I just want to understand what the notation means. --prime mover (talk) 11:34, 27 May 2022 (UTC)


 * OK, I thought if you try, then you realize the issue.
 * By the way, you have already encountered the same thing here Euclidean Space is Banach Space/Proof 2, please check it again. There, the outer index is the subscript and the inner is the superscript as opposite to here. Here I was forced to use $x^n$ for the outer index because the inner index $x_n$ is already defined. --Usagiop
 * Sorry, was that page one of mine? I don't remember reading it. --prime mover (talk) 13:48, 27 May 2022 (UTC)

Anyway I improve the proof with $x^{\paren n}$, after my networking issue is gone --Usagiop


 * Based on what was written here I jumped to a conclusion and added the clarifying dependent clause. Alternatively we could just work with $x_{n,i}$ but I was too lazy. &mdash; Lord_Farin (talk) 08:24, 28 May 2022 (UTC)


 * Sorry, but where it is now is back where we started, with obscure and misleading notation. Indeed, the two-dimensional subscript approach would be brilliant, it is so much easier to identify what is what. Writing it as $x_{n, i}$ makes so much more intrinsic sense. Then it can be made immediately apparent exactly what things are. --prime mover (talk) 10:39, 28 May 2022 (UTC)


 * Oh, and what else would help would be not using the same letter for both indices unless they genuinely need to be the same value on both dimensions. --prime mover (talk) 10:41, 28 May 2022 (UTC)


 * If you use $x_{n, i}\in\R$, how do you express the Cauchy sequence in $\R^\omega$ that is currently denoted by $\sequence {x^n} _{n \mathop \in \N}$? Does anyone have now a concrete proposal? --Usagiop


 * I'm going to agree that $\sequence {x^{(n)} }_{n \mathop \in \N}$, with parenthesis in the exponent, is the clearer notation here. To me it's clearer what's going on if you have $\norm {x^{(n)}_j - x^{(m)}_j}$ vs if you have $\norm {x_{n, j} - x_{m, j} }$. It's not unusual notation and I would be surprised if it doesn't appear elsewhere on the site to be honest. Caliburn (talk) 14:33, 28 May 2022 (UTC)