Martin Sleziak's room - 2016-08-09

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1:39 PM

Related to equivalence of $\|\cdot\|_\infty$ and $\|\cdot\|_1$: math.stackexchange.com/questions/25157/… — Martin Sleziak 8 hours ago

@MartinSleziak In fact , I do not care about the proof. What is the intuition behind the proof( theorem)? I know they are correct — stander Qiu 40 mins ago

Well, if you are only interested in the two norms ($\|\cdot\|_\infty$ and $\|\cdot\|_1$ it seems more-or-less obvious. The sum of $N$ positive elements is at most $N$-times the maximal of them. And you combine this with triangle inequality. The important thing is that there is only finitely many summands. — Martin Sleziak 31 mins ago

@MartinSleziak Assume that all the elements are nonnegative. the sum of all the elements is absolutely larger than the maximum. How could they be equivalent? — stander Qiu 9 mins ago

I am not sure what you mean. $\ell_1$ norm is defined as sum of absolute values. (BTW could you perhaps give a more precise reference than just the title of the book?) If needed, we can continue in this chat, so that we do not leave too many off-topic comments on your question. — Martin Sleziak 6 secs ago

Jan 24 '15 at 18:58, by Martin Sleziak

Maybe it will be useful to remind that instructions how to use MathJax in SE chat can be found here and here.

It seem to me that you are asking something along these lines:

$$\max |x_i| \le \sum_{i=1}^N |x_i| \tag{1}$$

$$\sum_{i=1}^N |x_i| \le N\cdot \max |x_i| \tag{2}$$

(1) is true simply because $\max|x_i|$ is one of the elements in the sum.

(2) is true because there are $N$ elements in the sum and each of them is less than or equal to $\max |x_i|$.

Hi~ The book is Matrix Analysis 2nd by Roger A. Horn

@standerQiu That would be probably worth mentioning in the quesiton.

I have Horn R.A., Johnson C.R. Matrix analysis (CUP, 1990) here. It seems to be reprinted first edition.

My question is : $l_1$ norm is defined as sum of absolute values, and infinite norm is the maximum

1:54 PM

We are talking about norms of vectors?

Page 320, just the bottom

Yep

Section 5.7. "Vector norms on matrices"?

Er, no. 5.2 Examples of norms and inner products

I see. It is page 320 in the second edition, I have the first edition.

Yep

1:56 PM

They define $\|x\|_1=|x_1|+\dots+|x_n|$, i.e. as the sum of absolute values.

And $\|x\|_\infty$ as maximum of absolute values.

Yep, this is the definition of the vector norms

Ok.

So I think that (1) and (2) I wrote above shows how they are equivalent.

One of the summands in the sum $\|x\|_1=|x_1|+\dots+|x_n|$ is equal to $\|x\|_\infty$. Which means that $\|x\|_\infty \le \|x\|_1$.

On the other hand, each of those summands is $\le \|x\|_\infty$. Therefore $$\|x\|_1=|x_1|+\dots+|x_n|\le n\cdot\|x\|_\infty.$$

I know how to proof( by k-norms). But just look at the situation where all the elements are nonnegative

The maximun = the sum. This is counter-intuitive and wrong.

That's not true.

The sum is n $\times$ maximum.

The sum is all the elements

2:04 PM

Yes.

And the norms are equivalent

What I wrote is wrong. It is not n$\times$ maximum.

@standerQiu And why is that a problem? Equivalent is not the same as equal.

What? Equivalent is not the same as equal. What does that mean?

The definition of equivalent norms is the there exist constants A and B such that $A\|x\|_1\le\|x\|_2\le B\|x\|_1$$ holds for every $x$.

You can find this in the Wikipedia article.

Oh. I used to mix the equivalent with the equal... sigh..

And what is the property of equivalent? Just for 1-norm and inf-norm

2:11 PM

The definition of equivalent norms is the there exist constants A and B such that $$A\|x\|_1\le\|x\|_2\le B\|x\|_1$$ holds for every $x$.

Or, in your case $$A\|x\|_1\le\|x\|_\infty\le B\|x\|_1.$$

At least that is the definition I am used to.

Er, I mean the intuition of equivalent? What do we know behind the definition( maths)

Horn and Johnson introduce equivalence of norms in Definition 5.4.7 on page 327.

In fact, their definition seems to be more intuitive. (Convergence in one norm is the same as convergence in the other one.)

0

Q: Significance of equivalent norms?

So two norms $|| \cdot ||_1$ and $|| \cdot ||_2$ are equivalent if $$\exists \ c \in \mathbb{R}$$ such that $\forall x \in X$ we have that $$\frac{1}{c}|| \cdot ||_1 \le || \cdot ||_2 \le c|| \cdot ||_1$$ What is the intuition I should be taking from this definition? Why is $|| \cdot ||_2$ equiv...

general-topology functional-analysis

Oh! I see! Thank you very much! I am so sorry for bothering you!

That's ok.

BTW, what do you use to type latex? How can you type in common

common characters and formulas in the same time

2:15 PM

I am not sure what you mean common?

LaTeX consists of common characters.

Do you just type in the textpad?

If you mean rendering formulas in chat, there is a bookmarklet you can use:

Jan 24 '15 at 18:58, by Martin Sleziak

Maybe it will be useful to remind that instructions how to use MathJax in SE chat can be found here and here.

@standerQiu At the moment I type just here in chat window. If I have a longer text, I use WinEdt.

8

A: LaTeX Editors/IDEs

Vim with Snipmate plugin and Rubber Available for: Windows, Mac, Linux and others Open Source I used to use Vim-Latex, but I found it too heavy-weight and rigid. Snipmate provides a subset of the functionality, but it is easier to customize and works for any programming language.

Oh, I see. Aha, I am not used to typing LaTeX without IDE..

62

A: TeX Community Polls

My favorite (La)TeX editor is...

Sorry, I will have to go away from the computer now.

Thanks !

2:18 PM

See you later and good luck with your studying!

See you!

2:29 PM

Hello

Er. Still a problem. The text book write inf norm = 1 norm. You mean that this is equivalent instead of euqal?

Hello

Kinda a weird question, but does anyone know of any math forum-type websites

where you can share discoveries/proofs, rather than questions?

I am not sure what you mean..

Hey@Polygon, I am sorry that I have to leave away from the lab.. I have something to do . See you!

bye

@standerQiu If you mean on page 320 below, that is $\|\cdot\|_\infty=\|\cdot\|_{[1]}$ and not $\|\cdot\|_\infty=\|\cdot\|_{1}$.

The equality $\|x\|_\infty=\|x\|_{[1]}$ is ok, because both of them mean the largest entry (in the absolute value).

@Polygon You might have better luck asking in the main chatroom. (More people will see your message.)

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