Dtjytyk

What's the point in a preamp?

How to read αἱμύλιος or when to aspirate

"... to apply for a visa" or "... and applied for a visa"?

My body leaves; my core can stay

Huge performance difference of the command find with and without using %M option to show permissions

Do warforged have souls?

University's motivation for having tenure-track positions

Identify 80s or 90s comics with ripped creatures (not dwarves)

Mortgage adviser recommends a longer term than necessary combined with overpayments

Windows 10: How to Lock (not sleep) laptop on lid close?

Didn't get enough time to take a Coding Test - what to do now?

Can we generate random numbers using irrational numbers like π and e?

Why doesn't a hydraulic lever violate conservation of energy?

How do you keep chess fun when your opponent constantly beats you?

"is" operation returns false even though two objects have same id

Why did Peik Lin say, "I'm not an animal"?

Example of compact Riemannian manifold with only one geodesic.

Keeping a retro style to sci-fi spaceships?

Why can't devices on different VLANs, but on the same subnet, communicate?

Can the DM override racial traits?

Is there a writing software that you can sort scenes like slides in PowerPoint?

Circular reasoning in L'Hopital's rule

Can withdrawing asylum be illegal?

how can a perfect fourth interval be considered either consonant or dissonant?

Doubt regarding Compactness argument used in Proof

2

2

$begingroup$

In that Book I come across following theorem.

Notation

Author mentioned that by standard argument of compact we can extract sequnce such that $A_{t_{k-1}^{-1}A_{t_k}}in V$.

I do not understand this standard argument .

I would be really thankful If someone help me out

Any Help will be appreciated

real-analysis proof-explanation

share|cite|improve this question

edited Mar 25 at 8:43

SRJ

asked Mar 22 at 13:16

SRJSRJ

1,8981620

$endgroup$

add a comment | 

2

2

$begingroup$

In that Book I come across following theorem.

Notation

Author mentioned that by standard argument of compact we can extract sequnce such that $A_{t_{k-1}^{-1}A_{t_k}}in V$.

I do not understand this standard argument .

I would be really thankful If someone help me out

Any Help will be appreciated

real-analysis proof-explanation

share|cite|improve this question

edited Mar 25 at 8:43

SRJ

asked Mar 22 at 13:16

SRJSRJ

1,8981620

$endgroup$

add a comment | 

$begingroup$

In that Book I come across following theorem.

Notation

Author mentioned that by standard argument of compact we can extract sequnce such that $A_{t_{k-1}^{-1}A_{t_k}}in V$.

I do not understand this standard argument .

I would be really thankful If someone help me out

Any Help will be appreciated

real-analysis proof-explanation

share|cite|improve this question

edited Mar 25 at 8:43

SRJ

asked Mar 22 at 13:16

SRJSRJ

1,8981620

$endgroup$

share|cite|improve this question

edited Mar 25 at 8:43

SRJ

asked Mar 22 at 13:16

SRJSRJ

1,8981620

share|cite|improve this question

edited Mar 25 at 8:43

SRJ

asked Mar 22 at 13:16

SRJSRJ

1,8981620

asked Mar 22 at 13:16

SRJSRJ

1,8981620

asked Mar 22 at 13:16

SRJSRJ

1,8981620

1 Answer
1

active

oldest

votes

2

$begingroup$

Hint: The operation $varphi:Gtimes Gto G, (A,B)mapsto A^{-1}B$ is continuous, hence so is $[0,1]times[0,1]to G, (t, s) mapsto A(t)^{-1}A(s)$.

Then the preimage of $V$ is open.

(You can also rely on a metric, if you wish, given by matrix norm.)

share|cite|improve this answer

answered Mar 22 at 14:32

BerciBerci

62k23776

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sir V is already given to be an open neighbourhood of I in G. I still do not understand. Can you please elaborate? Thanks a lot
  $endgroup$
  – SRJ
  Mar 23 at 10:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  As an open set, $varphi^{-1}(V)$ is a union of open balls, hence the diagonal ${(t,t):tin[0,1]}$ can be covered by finitely many of them. Let $varepsilon$ be the radius of the smallest ball among these, then $A(t)^{-1}A(s)in V$ whenever $|s-t|<varepsilon$.
  $endgroup$
  – Berci
  Mar 23 at 13:47
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks a lot... I required some time to digest argument. But Once I understand ... I became too happy AS it uses very basic idea and bypasses very High End tools....Once again thanks....
  $endgroup$
  – SRJ
  Mar 24 at 5:30
  

  
  
  
  

add a comment | 

Your Answer

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3158125%2fdoubt-regarding-compactness-argument-used-in-proof%23new-answer', 'question_page');
}
);

Post as a guest

Name

Email

Required, but never shown

2

$begingroup$

Hint: The operation $varphi:Gtimes Gto G, (A,B)mapsto A^{-1}B$ is continuous, hence so is $[0,1]times[0,1]to G, (t, s) mapsto A(t)^{-1}A(s)$.

Then the preimage of $V$ is open.

(You can also rely on a metric, if you wish, given by matrix norm.)

share|cite|improve this answer

answered Mar 22 at 14:32

BerciBerci

62k23776

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sir V is already given to be an open neighbourhood of I in G. I still do not understand. Can you please elaborate? Thanks a lot
  $endgroup$
  – SRJ
  Mar 23 at 10:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  As an open set, $varphi^{-1}(V)$ is a union of open balls, hence the diagonal ${(t,t):tin[0,1]}$ can be covered by finitely many of them. Let $varepsilon$ be the radius of the smallest ball among these, then $A(t)^{-1}A(s)in V$ whenever $|s-t|<varepsilon$.
  $endgroup$
  – Berci
  Mar 23 at 13:47
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks a lot... I required some time to digest argument. But Once I understand ... I became too happy AS it uses very basic idea and bypasses very High End tools....Once again thanks....
  $endgroup$
  – SRJ
  Mar 24 at 5:30
  

  
  
  
  

add a comment | 

2

$begingroup$

Hint: The operation $varphi:Gtimes Gto G, (A,B)mapsto A^{-1}B$ is continuous, hence so is $[0,1]times[0,1]to G, (t, s) mapsto A(t)^{-1}A(s)$.

Then the preimage of $V$ is open.

(You can also rely on a metric, if you wish, given by matrix norm.)

share|cite|improve this answer

answered Mar 22 at 14:32

BerciBerci

62k23776

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sir V is already given to be an open neighbourhood of I in G. I still do not understand. Can you please elaborate? Thanks a lot
  $endgroup$
  – SRJ
  Mar 23 at 10:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  As an open set, $varphi^{-1}(V)$ is a union of open balls, hence the diagonal ${(t,t):tin[0,1]}$ can be covered by finitely many of them. Let $varepsilon$ be the radius of the smallest ball among these, then $A(t)^{-1}A(s)in V$ whenever $|s-t|<varepsilon$.
  $endgroup$
  – Berci
  Mar 23 at 13:47
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks a lot... I required some time to digest argument. But Once I understand ... I became too happy AS it uses very basic idea and bypasses very High End tools....Once again thanks....
  $endgroup$
  – SRJ
  Mar 24 at 5:30
  

  
  
  
  

add a comment | 

$begingroup$

Hint: The operation $varphi:Gtimes Gto G, (A,B)mapsto A^{-1}B$ is continuous, hence so is $[0,1]times[0,1]to G, (t, s) mapsto A(t)^{-1}A(s)$.

Then the preimage of $V$ is open.

(You can also rely on a metric, if you wish, given by matrix norm.)

share|cite|improve this answer

answered Mar 22 at 14:32

BerciBerci

62k23776

$endgroup$

share|cite|improve this answer

answered Mar 22 at 14:32

BerciBerci

62k23776

answered Mar 22 at 14:32

BerciBerci

62k23776

answered Mar 22 at 14:32

BerciBerci

62k23776