I How are the following three definitions subtly different?

[elias001]

Questions;

For the three definitions quoted in the below Background section, I would like to know what the subtle differences are. I know they all have to do with extracting diagonal sequence from a bunch of sequences and its convergence. Especially with Definition 2., why it seems to be written so much different in its property $(b)$. I am guessing for the other two definitions, they are appear to be similar in wordings and notations, but I would like somebody who have seen these definitions or work with them explain to me their subtle differences and reasons for them.

Background

The following three definitions are taken respectively from the following books:

An Introduction to Multivariable Analysis from Vector to Manifold by: Piotr Mikusinski and Michael D. Taylor

Hilbert Spaces with Applications by Lokenath Debnath and Piotr Mikusinski

Operational Calculus, Vol 2 by: Jan Mikusinski, Thomas K. Boehme


[From Mikusinski and Taylor]

Definition 1.

Let $(X,d)$ be a metric space and let $x_{n,k} \in X$ for $k,n \in N$. Prove that if

$$x_{1,1},x_{1,2},x_{1,3}, \ldots \rightarrow x$$

$$x_{2,1},x_{2,2},x_{2,3}, \ldots \rightarrow x$$
$$x_{3,1},x_{3,2},x_{3,3}, \ldots \rightarrow x$$
$$\vdots$$
$$x_{n,1},x_{n,2},x_{n,3}, \ldots \rightarrow x$$
$$\vdots$$

then there exists an increasing sequence of natural numbers $p_n$ such that $x_{n,p_n} \rightarrow x.$

[From Debnath and Mikusinski]

Definition 2.

Theorem 1.4.12. (Diagonal theorem) Let $E$ be a normed space and let $(x_{ij}), i,j\in \Bbb{N}$ be an infinite matrix of elements of $E$. If

$(a)$ $\lim_{i\to\infty}=0$ for every $j\in \Bbb{N}$ and

$(b)$ every increasing sequence of indices $(p_j)$ has a subsequence $(q_j)$ such that
$$\lim_{i\to\infty}\sum_{j=1}^{\infty}x_{q_iq_j}=0,$$

then $\lim_{i\to\infty}x_{ij}=0.$

[From Mikusinski and Boehme]

Definition 3. 6. The diagonal subsequence property.

A space with a sequential convergence defined on it is said to have diagonal subsequence property if for each double sequence $x_{n,m}, n,m=1,2,\ldots,$ satisfying

$$(i)\quad\text{ for each }n\;\;\;x_{n,m}\to x_n\text{ as }m\to \infty,$$
and
$$(ii)\quad x_n\to x\text{ as }n\to \infty$$

there is a diagonal subsequence $p_i=x_{n(i),m(i)}$ where $n(i)\to \infty, m(i)\to \infty$ as $i\to \infty,$ such that
$$p_i\to x\text{ as }i\to \infty.$$

Thank you in advance.

 

[martinbn]

How are definitions 1 and 2 definitions? They are statements, one given as an exercise, the other a theorem. Also you mistyped something in "definition 2". The first limit, and what is $t$?

 

[elias001]

@martinbn thank you for noticing, I fixed it. I am not sure if definition 2 property (b) would make it more general, since there is a summation there where as the other two definitions, there are not.

 

[martinbn]

@martinbn thank you for noticing, I fixed it. I am not sure if definition 2 property (b) would make it more general, since there is a summation there where as the other two definitions, there are not.

Property a) in definition 2, the limit of what? And why do you call them definitions? Two is clearly a theorem, no?

 

[jedishrfu]

I am confused. They look like theorems to be proved or problems to be solved. I don't see any definitions to compare.

As an example:

A triangle is a plane figure that has three straight bounding sides.

 

[elias001]

@martinbn In all three definitions, I call it definitions for ease of reference, I meant why are they all phrased so slightly differently, especially in the case of #2 compare with the other two.

 

[elias001]

@jedishrfu all three are about the diagonal property of subsequences, but they are all phrased slightly differently. Are any of them a generalization of the other two?

 

[jedishrfu]

@martinbn In all three definitions, I call it definitions for ease of reference, I meant why are they all phrased so slightly differently, especially in the case of #2 compare with the other two.

Mathematics aims to be precise in its language, so avoid calling something a definition if it's not one.

You could instead say all three descriptions seem to be defined differently.

 

[elias001]

@jedishrfu ok, why does description #2 seem to involve a limit of a sum of sequences while description #1 and #3 both don't.

 

[FactChecker]

For the first two, by "definition" or "description" do you mean the requirements for applying the theorems 1 and 2? The third one is a legitimate definition.

It's not unusual to state things slightly differently even if the general situation is similar. There are a lot of theorems that can be proven true.

 

[elias001]

@FactChecker I am trying to understand why #2 property (b) is different than the other two descriptions/Definitions.

 

[martinbn]

Theorem 1.4.12. (Diagonal theorem) Let $E$ be a normed space and let $(x_{ij}), i,j\in \Bbb{N}$ be an infinite matrix of elements of $E$. If

$(a)$ $\lim_{i\to\infty}=0$ for every $j\in \Bbb{N}$ and

What's in the limit?

 

[elias001]

@martinbn it is $x_{ij}$. Thank you for noticing

 

[FactChecker]

@FactChecker I am trying to understand why #2 property (b) is different than the other two descriptions/Definitions.

Because they are trying to prove something slightly different.

 

[elias001]

@FactChecker All three descriptions has to do with diagonal subsequence property. Why is two of them formulated as a theorem, and the other one a definition.

 

[FactChecker]

@FactChecker All three descriptions has to do with diagonal subsequence property. Why is two of them formulated as a theorem, and the other one a definition.

Because two of them are proving a theorem and one of them is just defining the "diagonal subsequence property" without proving anything.
Notice that the two theorems do not state that the diagonal subsequence property holds. Maybe they are trying to prove that it does hold in that particular situation.