A Lipschitz Implicit Function Theorem.
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I look for a reference (book or article) that contains the statement of a version of the implicit function theorem as stated below. This statement I found in notes (with due proof) on the implicit function written by KC Border.
In these notes the author states that the proof is based on the proof of the implicit function theorem for two-variable functions given by Apostol in his calculus book. I looked at the proof given by Apostol in his calculus book (Calculus vol.2) and in fact the proof method of the above theorem is, with due adaptations, the same.
I am writing notes on the implicit function theorem. However, I did not want to have to repeat the proof, but to indicate to the reader a formal reference containing the proof.
Update 11/13/2018
I have already searched in the following textbooks
Implicit Functions and Solution Mappings. A View from Variational Analysis
The Implicit Function Theorem: History, Theory, and Applications
Nonlinear Functional Analysis
But I did not succeed.
real-analysis general-topology analysis reference-request implicit-function-theorem
asked Nov 10 at 13:42
MathOverview
8,51443063
add a comment |
up vote
4
down vote
favorite
I look for a reference (book or article) that contains the statement of a version of the implicit function theorem as stated below. This statement I found in notes (with due proof) on the implicit function written by KC Border.
In these notes the author states that the proof is based on the proof of the implicit function theorem for two-variable functions given by Apostol in his calculus book. I looked at the proof given by Apostol in his calculus book (Calculus vol.2) and in fact the proof method of the above theorem is, with due adaptations, the same.
I am writing notes on the implicit function theorem. However, I did not want to have to repeat the proof, but to indicate to the reader a formal reference containing the proof.
Update 11/13/2018
I have already searched in the following textbooks
Implicit Functions and Solution Mappings. A View from Variational Analysis
The Implicit Function Theorem: History, Theory, and Applications
Nonlinear Functional Analysis
But I did not succeed.
real-analysis general-topology analysis reference-request implicit-function-theorem
asked Nov 10 at 13:42
MathOverview
8,51443063
What does $x_{-}y$ mean?
– zhw.
Nov 22 at 21:40
@zhw. Here $ x $ and $ y $ are variables in $mathbb{R}$ and $x_{-}y$ is $ x-y $. The subtraction of $ x $ by $ y $.
– MathOverview
Nov 22 at 21:44
@zhw. I edited the image. I believe that there is no more doubt now.
– MathOverview
Nov 22 at 21:58
What is wrong with a reference to that pdf?
– HTFB
Nov 23 at 16:26
@HTFB The position of the minus sign in the expression $ y-x $ was not at the correct height. This seemed to indicate some abstract operation between $ x $ and $ and $ y$. But it has already been corrected.
– MathOverview
Nov 23 at 23:24
add a comment |
up vote
4
down vote
favorite
up vote
4
down vote
favorite
I look for a reference (book or article) that contains the statement of a version of the implicit function theorem as stated below. This statement I found in notes (with due proof) on the implicit function written by KC Border.
In these notes the author states that the proof is based on the proof of the implicit function theorem for two-variable functions given by Apostol in his calculus book. I looked at the proof given by Apostol in his calculus book (Calculus vol.2) and in fact the proof method of the above theorem is, with due adaptations, the same.
I am writing notes on the implicit function theorem. However, I did not want to have to repeat the proof, but to indicate to the reader a formal reference containing the proof.
Update 11/13/2018
I have already searched in the following textbooks
Implicit Functions and Solution Mappings. A View from Variational Analysis
The Implicit Function Theorem: History, Theory, and Applications
Nonlinear Functional Analysis
But I did not succeed.
real-analysis general-topology analysis reference-request implicit-function-theorem
asked Nov 10 at 13:42
MathOverview
8,51443063
I look for a reference (book or article) that contains the statement of a version of the implicit function theorem as stated below. This statement I found in notes (with due proof) on the implicit function written by KC Border.
In these notes the author states that the proof is based on the proof of the implicit function theorem for two-variable functions given by Apostol in his calculus book. I looked at the proof given by Apostol in his calculus book (Calculus vol.2) and in fact the proof method of the above theorem is, with due adaptations, the same.
I am writing notes on the implicit function theorem. However, I did not want to have to repeat the proof, but to indicate to the reader a formal reference containing the proof.
Update 11/13/2018
I have already searched in the following textbooks
Implicit Functions and Solution Mappings. A View from Variational Analysis
The Implicit Function Theorem: History, Theory, and Applications
Nonlinear Functional Analysis
But I did not succeed.
real-analysis general-topology analysis reference-request implicit-function-theorem
real-analysis general-topology analysis reference-request implicit-function-theorem
asked Nov 10 at 13:42
MathOverview
8,51443063
asked Nov 10 at 13:42
MathOverview
8,51443063
edited Nov 22 at 21:57
asked Nov 10 at 13:42
MathOverview
8,51443063
asked Nov 10 at 13:42
MathOverview
8,51443063
asked Nov 10 at 13:42
MathOverview
8,51443063
8,51443063
What does $x_{-}y$ mean?
– zhw.
Nov 22 at 21:40
@zhw. Here $ x $ and $ y $ are variables in $mathbb{R}$ and $x_{-}y$ is $ x-y $. The subtraction of $ x $ by $ y $.
– MathOverview
Nov 22 at 21:44
@zhw. I edited the image. I believe that there is no more doubt now.
– MathOverview
Nov 22 at 21:58
What is wrong with a reference to that pdf?
– HTFB
Nov 23 at 16:26
@HTFB The position of the minus sign in the expression $ y-x $ was not at the correct height. This seemed to indicate some abstract operation between $ x $ and $ and $ y$. But it has already been corrected.
– MathOverview
Nov 23 at 23:24
add a comment |
What does $x_{-}y$ mean?
– zhw.
Nov 22 at 21:40
@zhw. Here $ x $ and $ y $ are variables in $mathbb{R}$ and $x_{-}y$ is $ x-y $. The subtraction of $ x $ by $ y $.
– MathOverview
Nov 22 at 21:44
@zhw. I edited the image. I believe that there is no more doubt now.
– MathOverview
Nov 22 at 21:58
What is wrong with a reference to that pdf?
– HTFB
Nov 23 at 16:26
@HTFB The position of the minus sign in the expression $ y-x $ was not at the correct height. This seemed to indicate some abstract operation between $ x $ and $ and $ y$. But it has already been corrected.
– MathOverview
Nov 23 at 23:24
What does $x_{-}y$ mean?
– zhw.
Nov 22 at 21:40
What does $x_{-}y$ mean?
– zhw.
Nov 22 at 21:40
@zhw. Here $ x $ and $ y $ are variables in $mathbb{R}$ and $x_{-}y$ is $ x-y $. The subtraction of $ x $ by $ y $.
– MathOverview
Nov 22 at 21:44
@zhw. Here $ x $ and $ y $ are variables in $mathbb{R}$ and $x_{-}y$ is $ x-y $. The subtraction of $ x $ by $ y $.
– MathOverview
Nov 22 at 21:44
@zhw. I edited the image. I believe that there is no more doubt now.
– MathOverview
Nov 22 at 21:58
@zhw. I edited the image. I believe that there is no more doubt now.
– MathOverview
Nov 22 at 21:58
What is wrong with a reference to that pdf?
– HTFB
Nov 23 at 16:26
What is wrong with a reference to that pdf?
– HTFB
Nov 23 at 16:26
@HTFB The position of the minus sign in the expression $ y-x $ was not at the correct height. This seemed to indicate some abstract operation between $ x $ and $ and $ y$. But it has already been corrected.
– MathOverview
Nov 23 at 23:24
@HTFB The position of the minus sign in the expression $ y-x $ was not at the correct height. This seemed to indicate some abstract operation between $ x $ and $ and $ y$. But it has already been corrected.
– MathOverview
Nov 23 at 23:24
add a comment |
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What does $x_{-}y$ mean?
– zhw.
Nov 22 at 21:40
@zhw. Here $ x $ and $ y $ are variables in $mathbb{R}$ and $x_{-}y$ is $ x-y $. The subtraction of $ x $ by $ y $.
– MathOverview
Nov 22 at 21:44
@zhw. I edited the image. I believe that there is no more doubt now.
– MathOverview
Nov 22 at 21:58
What is wrong with a reference to that pdf?
– HTFB
Nov 23 at 16:26
@HTFB The position of the minus sign in the expression $ y-x $ was not at the correct height. This seemed to indicate some abstract operation between $ x $ and $ and $ y$. But it has already been corrected.
– MathOverview
Nov 23 at 23:24