Showing that a given function is Frechet differentiable
$begingroup$
I'd like to do the following exercise from my book. The statement is as follows.
Let $gamma epsilon { C }^{ 1 }(Rtimes { R }^{ n })$,$c>0$ and ${ gamma }_{ 0 }epsilon { L }^{ 1 }({ R }^{ n })$ such that $left| { gamma }(z,x) right| le { gamma }_{ 0 }(x)+cleft| z right| $.
And let $G(f);=int _{ { R }^{ n } }^{ }{ { gamma }(f(x),x)dx } $
Show that $G({ L }^{ 1 }({ R }^{ n }))rightarrow R$ is Frechet differentiable and calculate the derivative.
It's the first exercise for Frechet differentiability after it was introduced so im not very fermiliar with it.
We need to show that $lim _{ hrightarrow 0 }{ frac { left| G(u+h)-G(u)-G'(u)h right| }{ { left| h right| }_{ { L }^{ 1 } } } } =0$
I started looking at the numerator so we have
$left| G(u+h)-G(u)-G'(u)h right| =int _{ { R }^{ n } }^{ }{ gamma (u(x)+h(x),x)-gamma (u(x),x)+gamma (u(x),x)h(x)dx } $
=$iint _{ u(x)+h(x) }^{ u(x) }{ gamma (s,x)ds-gamma (u(x),x)h(x)dx} $ =$iint _{ u(x)+h(x) }^{ u(x) }{ gamma (s,x)-gamma (u(x),x)dsdx } $
but how can i now make this expression smaller than an arbitrary $varepsilon >0$? is this calculation even usefull? And how can i use the Assumtion for $gamma $?
Any Help is greatly appreciated Thank you
real-analysis frechet-derivative
asked Dec 6 '18 at 21:09
MasterPIMasterPI
21818
$endgroup$
add a comment |
$begingroup$
I'd like to do the following exercise from my book. The statement is as follows.
Let $gamma epsilon { C }^{ 1 }(Rtimes { R }^{ n })$,$c>0$ and ${ gamma }_{ 0 }epsilon { L }^{ 1 }({ R }^{ n })$ such that $left| { gamma }(z,x) right| le { gamma }_{ 0 }(x)+cleft| z right| $.
And let $G(f);=int _{ { R }^{ n } }^{ }{ { gamma }(f(x),x)dx } $
Show that $G({ L }^{ 1 }({ R }^{ n }))rightarrow R$ is Frechet differentiable and calculate the derivative.
It's the first exercise for Frechet differentiability after it was introduced so im not very fermiliar with it.
We need to show that $lim _{ hrightarrow 0 }{ frac { left| G(u+h)-G(u)-G'(u)h right| }{ { left| h right| }_{ { L }^{ 1 } } } } =0$
I started looking at the numerator so we have
$left| G(u+h)-G(u)-G'(u)h right| =int _{ { R }^{ n } }^{ }{ gamma (u(x)+h(x),x)-gamma (u(x),x)+gamma (u(x),x)h(x)dx } $
=$iint _{ u(x)+h(x) }^{ u(x) }{ gamma (s,x)ds-gamma (u(x),x)h(x)dx} $ =$iint _{ u(x)+h(x) }^{ u(x) }{ gamma (s,x)-gamma (u(x),x)dsdx } $
but how can i now make this expression smaller than an arbitrary $varepsilon >0$? is this calculation even usefull? And how can i use the Assumtion for $gamma $?
Any Help is greatly appreciated Thank you
real-analysis frechet-derivative
asked Dec 6 '18 at 21:09
MasterPIMasterPI
21818
$endgroup$
$begingroup$
Your calculations are completely irrelevant to this exercise. I suggest you make sure you understand the definition of the Frechet derivative, and after that look for an example for the computation of a Frechet derivative of some operator (should appear in any textbook dealing with that subject).
$endgroup$
– MOMO
Dec 6 '18 at 23:32
$begingroup$
Moreover, I recommend you try and understand why the equality between integrals you wrote are incorrect, although it is not needed to the exercise.
$endgroup$
– MOMO
Dec 6 '18 at 23:37
$begingroup$
I tried the exercise for well over an hour, I think I know what the derivative is but I couldn't prove it. There is an issue with regards of the vector spaces. I think you need to assume $|partial_z gamma|$ is bounded and then the exercise would follow in a rather straight-forward way using derivative rules (chain rule, mostly).
$endgroup$
– Will M.
Dec 8 '18 at 6:09
add a comment |
$begingroup$
I'd like to do the following exercise from my book. The statement is as follows.
Let $gamma epsilon { C }^{ 1 }(Rtimes { R }^{ n })$,$c>0$ and ${ gamma }_{ 0 }epsilon { L }^{ 1 }({ R }^{ n })$ such that $left| { gamma }(z,x) right| le { gamma }_{ 0 }(x)+cleft| z right| $.
And let $G(f);=int _{ { R }^{ n } }^{ }{ { gamma }(f(x),x)dx } $
Show that $G({ L }^{ 1 }({ R }^{ n }))rightarrow R$ is Frechet differentiable and calculate the derivative.
It's the first exercise for Frechet differentiability after it was introduced so im not very fermiliar with it.
We need to show that $lim _{ hrightarrow 0 }{ frac { left| G(u+h)-G(u)-G'(u)h right| }{ { left| h right| }_{ { L }^{ 1 } } } } =0$
I started looking at the numerator so we have
$left| G(u+h)-G(u)-G'(u)h right| =int _{ { R }^{ n } }^{ }{ gamma (u(x)+h(x),x)-gamma (u(x),x)+gamma (u(x),x)h(x)dx } $
=$iint _{ u(x)+h(x) }^{ u(x) }{ gamma (s,x)ds-gamma (u(x),x)h(x)dx} $ =$iint _{ u(x)+h(x) }^{ u(x) }{ gamma (s,x)-gamma (u(x),x)dsdx } $
but how can i now make this expression smaller than an arbitrary $varepsilon >0$? is this calculation even usefull? And how can i use the Assumtion for $gamma $?
Any Help is greatly appreciated Thank you
real-analysis frechet-derivative
asked Dec 6 '18 at 21:09
MasterPIMasterPI
21818
$endgroup$
I'd like to do the following exercise from my book. The statement is as follows.
Let $gamma epsilon { C }^{ 1 }(Rtimes { R }^{ n })$,$c>0$ and ${ gamma }_{ 0 }epsilon { L }^{ 1 }({ R }^{ n })$ such that $left| { gamma }(z,x) right| le { gamma }_{ 0 }(x)+cleft| z right| $.
And let $G(f);=int _{ { R }^{ n } }^{ }{ { gamma }(f(x),x)dx } $
Show that $G({ L }^{ 1 }({ R }^{ n }))rightarrow R$ is Frechet differentiable and calculate the derivative.
It's the first exercise for Frechet differentiability after it was introduced so im not very fermiliar with it.
We need to show that $lim _{ hrightarrow 0 }{ frac { left| G(u+h)-G(u)-G'(u)h right| }{ { left| h right| }_{ { L }^{ 1 } } } } =0$
I started looking at the numerator so we have
$left| G(u+h)-G(u)-G'(u)h right| =int _{ { R }^{ n } }^{ }{ gamma (u(x)+h(x),x)-gamma (u(x),x)+gamma (u(x),x)h(x)dx } $
=$iint _{ u(x)+h(x) }^{ u(x) }{ gamma (s,x)ds-gamma (u(x),x)h(x)dx} $ =$iint _{ u(x)+h(x) }^{ u(x) }{ gamma (s,x)-gamma (u(x),x)dsdx } $
but how can i now make this expression smaller than an arbitrary $varepsilon >0$? is this calculation even usefull? And how can i use the Assumtion for $gamma $?
Any Help is greatly appreciated Thank you
real-analysis frechet-derivative
real-analysis frechet-derivative
asked Dec 6 '18 at 21:09
MasterPIMasterPI
21818
asked Dec 6 '18 at 21:09
MasterPIMasterPI
21818
asked Dec 6 '18 at 21:09
MasterPIMasterPI
21818
asked Dec 6 '18 at 21:09
MasterPIMasterPI
21818
asked Dec 6 '18 at 21:09
MasterPIMasterPI
21818
21818
$begingroup$
Your calculations are completely irrelevant to this exercise. I suggest you make sure you understand the definition of the Frechet derivative, and after that look for an example for the computation of a Frechet derivative of some operator (should appear in any textbook dealing with that subject).
$endgroup$
– MOMO
Dec 6 '18 at 23:32
$begingroup$
Moreover, I recommend you try and understand why the equality between integrals you wrote are incorrect, although it is not needed to the exercise.
$endgroup$
– MOMO
Dec 6 '18 at 23:37
$begingroup$
I tried the exercise for well over an hour, I think I know what the derivative is but I couldn't prove it. There is an issue with regards of the vector spaces. I think you need to assume $|partial_z gamma|$ is bounded and then the exercise would follow in a rather straight-forward way using derivative rules (chain rule, mostly).
$endgroup$
– Will M.
Dec 8 '18 at 6:09
add a comment |
$begingroup$
Your calculations are completely irrelevant to this exercise. I suggest you make sure you understand the definition of the Frechet derivative, and after that look for an example for the computation of a Frechet derivative of some operator (should appear in any textbook dealing with that subject).
$endgroup$
– MOMO
Dec 6 '18 at 23:32
$begingroup$
Moreover, I recommend you try and understand why the equality between integrals you wrote are incorrect, although it is not needed to the exercise.
$endgroup$
– MOMO
Dec 6 '18 at 23:37
$begingroup$
I tried the exercise for well over an hour, I think I know what the derivative is but I couldn't prove it. There is an issue with regards of the vector spaces. I think you need to assume $|partial_z gamma|$ is bounded and then the exercise would follow in a rather straight-forward way using derivative rules (chain rule, mostly).
$endgroup$
– Will M.
Dec 8 '18 at 6:09
$begingroup$
Your calculations are completely irrelevant to this exercise. I suggest you make sure you understand the definition of the Frechet derivative, and after that look for an example for the computation of a Frechet derivative of some operator (should appear in any textbook dealing with that subject).
$endgroup$
– MOMO
Dec 6 '18 at 23:32
$begingroup$
Your calculations are completely irrelevant to this exercise. I suggest you make sure you understand the definition of the Frechet derivative, and after that look for an example for the computation of a Frechet derivative of some operator (should appear in any textbook dealing with that subject).
$endgroup$
– MOMO
Dec 6 '18 at 23:32
$begingroup$
Moreover, I recommend you try and understand why the equality between integrals you wrote are incorrect, although it is not needed to the exercise.
$endgroup$
– MOMO
Dec 6 '18 at 23:37
$begingroup$
Moreover, I recommend you try and understand why the equality between integrals you wrote are incorrect, although it is not needed to the exercise.
$endgroup$
– MOMO
Dec 6 '18 at 23:37
$begingroup$
I tried the exercise for well over an hour, I think I know what the derivative is but I couldn't prove it. There is an issue with regards of the vector spaces. I think you need to assume $|partial_z gamma|$ is bounded and then the exercise would follow in a rather straight-forward way using derivative rules (chain rule, mostly).
$endgroup$
– Will M.
Dec 8 '18 at 6:09
$begingroup$
I tried the exercise for well over an hour, I think I know what the derivative is but I couldn't prove it. There is an issue with regards of the vector spaces. I think you need to assume $|partial_z gamma|$ is bounded and then the exercise would follow in a rather straight-forward way using derivative rules (chain rule, mostly).
$endgroup$
– Will M.
Dec 8 '18 at 6:09
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
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
});
}
});
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3029053%2fshowing-that-a-given-function-is-frechet-differentiable%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3029053%2fshowing-that-a-given-function-is-frechet-differentiable%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
Your calculations are completely irrelevant to this exercise. I suggest you make sure you understand the definition of the Frechet derivative, and after that look for an example for the computation of a Frechet derivative of some operator (should appear in any textbook dealing with that subject).
$endgroup$
– MOMO
Dec 6 '18 at 23:32
$begingroup$
Moreover, I recommend you try and understand why the equality between integrals you wrote are incorrect, although it is not needed to the exercise.
$endgroup$
– MOMO
Dec 6 '18 at 23:37
$begingroup$
I tried the exercise for well over an hour, I think I know what the derivative is but I couldn't prove it. There is an issue with regards of the vector spaces. I think you need to assume $|partial_z gamma|$ is bounded and then the exercise would follow in a rather straight-forward way using derivative rules (chain rule, mostly).
$endgroup$
– Will M.
Dec 8 '18 at 6:09