A question on max norm
This is question 30 in chapter 7 from royden
"Let ${f_n}$ be a sequence in C[a, b] and $sum a_k$ a convergent series of positive numbers such
that $|| f_{k+1}- f_k||_{max} leq a_k, forall k$
Prove that
$|f_{k+n}(x)- f_k(x) | leq || f_{k+n}- f_k||_{max} leq sum_{j=n}^{infty}a_j , forall k,n $
Conclude that there is a function f$in$ C[a, b] such that ${fn} rightarrow f$ uniformly on [a, b] "
I know that, it is easy to solve this question if the series above starts from j=k , by adding and subtracting $f_{k+1},..., f_{k+n-1 } $ to the leftside and useing the triangular inequality.
But in this problem the seris starts from j=n.
I tried hard to solve it., however I have not any good idea.
Can you help me??
Any suggestion will be appreciated.
real-analysis linear-algebra functional-analysis measure-theory
edited yesterday
Davide Giraudo
125k16150260
asked yesterday
Mano.kiko
61
New contributor
Mano.kiko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
This is question 30 in chapter 7 from royden
"Let ${f_n}$ be a sequence in C[a, b] and $sum a_k$ a convergent series of positive numbers such
that $|| f_{k+1}- f_k||_{max} leq a_k, forall k$
Prove that
$|f_{k+n}(x)- f_k(x) | leq || f_{k+n}- f_k||_{max} leq sum_{j=n}^{infty}a_j , forall k,n $
Conclude that there is a function f$in$ C[a, b] such that ${fn} rightarrow f$ uniformly on [a, b] "
I know that, it is easy to solve this question if the series above starts from j=k , by adding and subtracting $f_{k+1},..., f_{k+n-1 } $ to the leftside and useing the triangular inequality.
But in this problem the seris starts from j=n.
I tried hard to solve it., however I have not any good idea.
Can you help me??
Any suggestion will be appreciated.
real-analysis linear-algebra functional-analysis measure-theory
edited yesterday
Davide Giraudo
125k16150260
asked yesterday
Mano.kiko
61
New contributor
Mano.kiko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
This seems like a typo to me. What you say you can prove ought to be the indended meaning.
– quid♦
yesterday
1
Well, the statement is not true and that's why you couldn't solve it. It looks like a typo. To see this, suppose it's true and take $n$ to $infty$.Then we get $lim_n f_n = f_k$ for all $k$ which is absurd.
– Song
yesterday
1
To elaborate slightly on my first comment, as a "test" consider constant functions $f_k$ defined via $f_{k+1}(x) = f_k(x)+a_k$.
– quid♦
yesterday
Thank you so much
– Mano.kiko
yesterday
add a comment |
This is question 30 in chapter 7 from royden
"Let ${f_n}$ be a sequence in C[a, b] and $sum a_k$ a convergent series of positive numbers such
that $|| f_{k+1}- f_k||_{max} leq a_k, forall k$
Prove that
$|f_{k+n}(x)- f_k(x) | leq || f_{k+n}- f_k||_{max} leq sum_{j=n}^{infty}a_j , forall k,n $
Conclude that there is a function f$in$ C[a, b] such that ${fn} rightarrow f$ uniformly on [a, b] "
I know that, it is easy to solve this question if the series above starts from j=k , by adding and subtracting $f_{k+1},..., f_{k+n-1 } $ to the leftside and useing the triangular inequality.
But in this problem the seris starts from j=n.
I tried hard to solve it., however I have not any good idea.
Can you help me??
Any suggestion will be appreciated.
real-analysis linear-algebra functional-analysis measure-theory
edited yesterday
Davide Giraudo
125k16150260
asked yesterday
Mano.kiko
61
New contributor
Mano.kiko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
This is question 30 in chapter 7 from royden
"Let ${f_n}$ be a sequence in C[a, b] and $sum a_k$ a convergent series of positive numbers such
that $|| f_{k+1}- f_k||_{max} leq a_k, forall k$
Prove that
$|f_{k+n}(x)- f_k(x) | leq || f_{k+n}- f_k||_{max} leq sum_{j=n}^{infty}a_j , forall k,n $
Conclude that there is a function f$in$ C[a, b] such that ${fn} rightarrow f$ uniformly on [a, b] "
I know that, it is easy to solve this question if the series above starts from j=k , by adding and subtracting $f_{k+1},..., f_{k+n-1 } $ to the leftside and useing the triangular inequality.
But in this problem the seris starts from j=n.
I tried hard to solve it., however I have not any good idea.
Can you help me??
Any suggestion will be appreciated.
real-analysis linear-algebra functional-analysis measure-theory
real-analysis linear-algebra functional-analysis measure-theory
edited yesterday
Davide Giraudo
125k16150260
asked yesterday
Mano.kiko
61
New contributor
Mano.kiko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited yesterday
Davide Giraudo
125k16150260
asked yesterday
Mano.kiko
61
New contributor
Mano.kiko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited yesterday
Davide Giraudo
125k16150260
edited yesterday
Davide Giraudo
125k16150260
edited yesterday
Davide Giraudo
125k16150260
125k16150260
asked yesterday
Mano.kiko
61
New contributor
Mano.kiko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
Mano.kiko
61
asked yesterday
Mano.kiko
61
61
New contributor
Mano.kiko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Mano.kiko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Mano.kiko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
This seems like a typo to me. What you say you can prove ought to be the indended meaning.
– quid♦
yesterday
1
Well, the statement is not true and that's why you couldn't solve it. It looks like a typo. To see this, suppose it's true and take $n$ to $infty$.Then we get $lim_n f_n = f_k$ for all $k$ which is absurd.
– Song
yesterday
1
To elaborate slightly on my first comment, as a "test" consider constant functions $f_k$ defined via $f_{k+1}(x) = f_k(x)+a_k$.
– quid♦
yesterday
Thank you so much
– Mano.kiko
yesterday
add a comment |
This seems like a typo to me. What you say you can prove ought to be the indended meaning.
– quid♦
yesterday
1
Well, the statement is not true and that's why you couldn't solve it. It looks like a typo. To see this, suppose it's true and take $n$ to $infty$.Then we get $lim_n f_n = f_k$ for all $k$ which is absurd.
– Song
yesterday
1
To elaborate slightly on my first comment, as a "test" consider constant functions $f_k$ defined via $f_{k+1}(x) = f_k(x)+a_k$.
– quid♦
yesterday
Thank you so much
– Mano.kiko
yesterday
This seems like a typo to me. What you say you can prove ought to be the indended meaning.
– quid♦
yesterday
This seems like a typo to me. What you say you can prove ought to be the indended meaning.
– quid♦
yesterday
1
1
Well, the statement is not true and that's why you couldn't solve it. It looks like a typo. To see this, suppose it's true and take $n$ to $infty$.Then we get $lim_n f_n = f_k$ for all $k$ which is absurd.
– Song
yesterday
Well, the statement is not true and that's why you couldn't solve it. It looks like a typo. To see this, suppose it's true and take $n$ to $infty$.Then we get $lim_n f_n = f_k$ for all $k$ which is absurd.
– Song
yesterday
1
1
To elaborate slightly on my first comment, as a "test" consider constant functions $f_k$ defined via $f_{k+1}(x) = f_k(x)+a_k$.
– quid♦
yesterday
To elaborate slightly on my first comment, as a "test" consider constant functions $f_k$ defined via $f_{k+1}(x) = f_k(x)+a_k$.
– quid♦
yesterday
Thank you so much
– Mano.kiko
yesterday
Thank you so much
– Mano.kiko
yesterday
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
});
}
});
Mano.kiko is a new contributor. Be nice, and check out our Code of Conduct.
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%2f3060656%2fa-question-on-max-norm%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
Mano.kiko is a new contributor. Be nice, and check out our Code of Conduct.
Mano.kiko is a new contributor. Be nice, and check out our Code of Conduct.
Mano.kiko is a new contributor. Be nice, and check out our Code of Conduct.
Mano.kiko is a new contributor. Be nice, and check out our Code of Conduct.
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3060656%2fa-question-on-max-norm%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
This seems like a typo to me. What you say you can prove ought to be the indended meaning.
– quid♦
yesterday
1
Well, the statement is not true and that's why you couldn't solve it. It looks like a typo. To see this, suppose it's true and take $n$ to $infty$.Then we get $lim_n f_n = f_k$ for all $k$ which is absurd.
– Song
yesterday
1
To elaborate slightly on my first comment, as a "test" consider constant functions $f_k$ defined via $f_{k+1}(x) = f_k(x)+a_k$.
– quid♦
yesterday
Thank you so much
– Mano.kiko
yesterday