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Isometry isomorphism in metric space [closed]

0

$begingroup$

Let $X$ be a complete metric space and $Y$ a subset of $X$. Prove that the completion of $Y$ is isometrically isomorphic to $overlineY$

Notice that $overlineY$ is not the completion of $Y$

I got the hint that I should prove there's an isometry from $Y$ to $overlineY$, and then the range of this isometry is dense, and in the end prove that $overlineY$ is complete.

I'm not fully understanding $overlineY$, and I have no idea where to start.
Help plz

analysis metric-spaces isometry

share|cite|improve this question

asked Mar 26 at 6:24

yyyyyyyy

82

$endgroup$

closed as off-topic by uniquesolution, José Carlos Santos, Song, Strants, dantopa Mar 26 at 22:22

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – uniquesolution, José Carlos Santos, Song, Strants, dantopa

If this question can be reworded to fit the rules in the help center, please edit the question.

add a comment | 

0

$begingroup$

Let $X$ be a complete metric space and $Y$ a subset of $X$. Prove that the completion of $Y$ is isometrically isomorphic to $overlineY$

Notice that $overlineY$ is not the completion of $Y$

I got the hint that I should prove there's an isometry from $Y$ to $overlineY$, and then the range of this isometry is dense, and in the end prove that $overlineY$ is complete.

I'm not fully understanding $overlineY$, and I have no idea where to start.
Help plz

analysis metric-spaces isometry

share|cite|improve this question

asked Mar 26 at 6:24

yyyyyyyy

82

$endgroup$

closed as off-topic by uniquesolution, José Carlos Santos, Song, Strants, dantopa Mar 26 at 22:22

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – uniquesolution, José Carlos Santos, Song, Strants, dantopa

If this question can be reworded to fit the rules in the help center, please edit the question.

add a comment | 

$begingroup$

Let $X$ be a complete metric space and $Y$ a subset of $X$. Prove that the completion of $Y$ is isometrically isomorphic to $overlineY$

Notice that $overlineY$ is not the completion of $Y$

I got the hint that I should prove there's an isometry from $Y$ to $overlineY$, and then the range of this isometry is dense, and in the end prove that $overlineY$ is complete.

I'm not fully understanding $overlineY$, and I have no idea where to start.
Help plz

analysis metric-spaces isometry

share|cite|improve this question

asked Mar 26 at 6:24

yyyyyyyy

82

$endgroup$

share|cite|improve this question

asked Mar 26 at 6:24

yyyyyyyy

82

share|cite|improve this question

asked Mar 26 at 6:24

yyyyyyyy

82

asked Mar 26 at 6:24

yyyyyyyy

82

asked Mar 26 at 6:24

yyyyyyyy

82

1 Answer
1

active

oldest

votes

1

$begingroup$

$overset - Y$ is the closure of $Y$ in $X$. Hint: if $y in overset - Y$ then there is a sequence $y_n$ in $Y$ converging to $y$. This sequence is a Cauchy sequence so there is a point $y'$ in the completion $Y'$ of $Y$ corresponding to this sequence. You have to just verify using definiitions that the map $y to y'$ is an isometric isomorphism.

share|cite|improve this answer

answered Mar 26 at 6:27

Kavi Rama MurthyKavi Rama Murthy

74.9k53270

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  why is there a $y'$ corresponding to $y_n$?
  $endgroup$
  – yyyy
  Mar 26 at 6:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Are familiar with the construction of the completion of a metric space?
  $endgroup$
  – Kavi Rama Murthy
  Mar 26 at 6:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See en.wikipedia.org/wiki/Complete_metric_space
  $endgroup$
  – Kavi Rama Murthy
  Mar 26 at 6:34
  

  
  
  
  

add a comment | 

1

$begingroup$

$overset - Y$ is the closure of $Y$ in $X$. Hint: if $y in overset - Y$ then there is a sequence $y_n$ in $Y$ converging to $y$. This sequence is a Cauchy sequence so there is a point $y'$ in the completion $Y'$ of $Y$ corresponding to this sequence. You have to just verify using definiitions that the map $y to y'$ is an isometric isomorphism.

share|cite|improve this answer

answered Mar 26 at 6:27

Kavi Rama MurthyKavi Rama Murthy

74.9k53270

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  why is there a $y'$ corresponding to $y_n$?
  $endgroup$
  – yyyy
  Mar 26 at 6:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Are familiar with the construction of the completion of a metric space?
  $endgroup$
  – Kavi Rama Murthy
  Mar 26 at 6:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See en.wikipedia.org/wiki/Complete_metric_space
  $endgroup$
  – Kavi Rama Murthy
  Mar 26 at 6:34
  

  
  
  
  

add a comment | 

1

$begingroup$

$overset - Y$ is the closure of $Y$ in $X$. Hint: if $y in overset - Y$ then there is a sequence $y_n$ in $Y$ converging to $y$. This sequence is a Cauchy sequence so there is a point $y'$ in the completion $Y'$ of $Y$ corresponding to this sequence. You have to just verify using definiitions that the map $y to y'$ is an isometric isomorphism.

share|cite|improve this answer

answered Mar 26 at 6:27

Kavi Rama MurthyKavi Rama Murthy

74.9k53270

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  why is there a $y'$ corresponding to $y_n$?
  $endgroup$
  – yyyy
  Mar 26 at 6:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Are familiar with the construction of the completion of a metric space?
  $endgroup$
  – Kavi Rama Murthy
  Mar 26 at 6:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See en.wikipedia.org/wiki/Complete_metric_space
  $endgroup$
  – Kavi Rama Murthy
  Mar 26 at 6:34
  

  
  
  
  

add a comment | 

$begingroup$

$overset - Y$ is the closure of $Y$ in $X$. Hint: if $y in overset - Y$ then there is a sequence $y_n$ in $Y$ converging to $y$. This sequence is a Cauchy sequence so there is a point $y'$ in the completion $Y'$ of $Y$ corresponding to this sequence. You have to just verify using definiitions that the map $y to y'$ is an isometric isomorphism.

share|cite|improve this answer

answered Mar 26 at 6:27

Kavi Rama MurthyKavi Rama Murthy

74.9k53270

$endgroup$

share|cite|improve this answer

answered Mar 26 at 6:27

Kavi Rama MurthyKavi Rama Murthy

74.9k53270

answered Mar 26 at 6:27

Kavi Rama MurthyKavi Rama Murthy

74.9k53270

answered Mar 26 at 6:27

Kavi Rama MurthyKavi Rama Murthy

74.9k53270