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. — 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)