Mixing
Find the norm of a linear functional in $L^2[0,1]$.
$begingroup$
Define the linear functional $f : L^2[0,1] text{(As completion of $C[0,1]$, all the continuous complex-valued function )} mapsto mathbb{C}$ by
$$f(psi)=3int_{0}^{1}psi(t)dt + iint_{0}^{1} psi(t)sin(pi t)cos(pi t)dt.$$
I have made use of Cauchy Schwarz to set a bound of $|f| leq 3+ frac{1}{sqrt{8}}$, yet I can not figure out what to substitute to obtain the norm. Can I get a small hint? Thanks!
functional-analysis norm
asked May 20 '16 at 10:10
Kenneth NgKenneth Ng
455
$endgroup$
add a comment |
$begingroup$
Define the linear functional $f : L^2[0,1] text{(As completion of $C[0,1]$, all the continuous complex-valued function )} mapsto mathbb{C}$ by
$$f(psi)=3int_{0}^{1}psi(t)dt + iint_{0}^{1} psi(t)sin(pi t)cos(pi t)dt.$$
I have made use of Cauchy Schwarz to set a bound of $|f| leq 3+ frac{1}{sqrt{8}}$, yet I can not figure out what to substitute to obtain the norm. Can I get a small hint? Thanks!
functional-analysis norm
asked May 20 '16 at 10:10
Kenneth NgKenneth Ng
455
$endgroup$
1
$begingroup$
Try writing it as $f(psi) = langle psi, grangle$ for some function $g$. Then find $|g|_2$.
$endgroup$
– Prahlad Vaidyanathan
May 20 '16 at 10:35
$begingroup$
always forget such representation.. thanks and solved ;)
$endgroup$
– Kenneth Ng
May 20 '16 at 10:51
1
$begingroup$
@KennethNg: In this case you might want to add your solution as an answer for further reference.
$endgroup$
– gerw
May 20 '16 at 11:23
add a comment |
$begingroup$
Define the linear functional $f : L^2[0,1] text{(As completion of $C[0,1]$, all the continuous complex-valued function )} mapsto mathbb{C}$ by
$$f(psi)=3int_{0}^{1}psi(t)dt + iint_{0}^{1} psi(t)sin(pi t)cos(pi t)dt.$$
I have made use of Cauchy Schwarz to set a bound of $|f| leq 3+ frac{1}{sqrt{8}}$, yet I can not figure out what to substitute to obtain the norm. Can I get a small hint? Thanks!
functional-analysis norm
asked May 20 '16 at 10:10
Kenneth NgKenneth Ng
455
$endgroup$
Define the linear functional $f : L^2[0,1] text{(As completion of $C[0,1]$, all the continuous complex-valued function )} mapsto mathbb{C}$ by
$$f(psi)=3int_{0}^{1}psi(t)dt + iint_{0}^{1} psi(t)sin(pi t)cos(pi t)dt.$$
I have made use of Cauchy Schwarz to set a bound of $|f| leq 3+ frac{1}{sqrt{8}}$, yet I can not figure out what to substitute to obtain the norm. Can I get a small hint? Thanks!
functional-analysis norm
functional-analysis norm
asked May 20 '16 at 10:10
Kenneth NgKenneth Ng
455
asked May 20 '16 at 10:10
Kenneth NgKenneth Ng
455
asked May 20 '16 at 10:10
Kenneth NgKenneth Ng
455
asked May 20 '16 at 10:10
Kenneth NgKenneth Ng
455
asked May 20 '16 at 10:10
Kenneth NgKenneth Ng
455
455
1
$begingroup$
Try writing it as $f(psi) = langle psi, grangle$ for some function $g$. Then find $|g|_2$.
$endgroup$
– Prahlad Vaidyanathan
May 20 '16 at 10:35
$begingroup$
always forget such representation.. thanks and solved ;)
$endgroup$
– Kenneth Ng
May 20 '16 at 10:51
1
$begingroup$
@KennethNg: In this case you might want to add your solution as an answer for further reference.
$endgroup$
– gerw
May 20 '16 at 11:23
add a comment |
1
$begingroup$
Try writing it as $f(psi) = langle psi, grangle$ for some function $g$. Then find $|g|_2$.
$endgroup$
– Prahlad Vaidyanathan
May 20 '16 at 10:35
$begingroup$
always forget such representation.. thanks and solved ;)
$endgroup$
– Kenneth Ng
May 20 '16 at 10:51
1
$begingroup$
@KennethNg: In this case you might want to add your solution as an answer for further reference.
$endgroup$
– gerw
May 20 '16 at 11:23
1
1
$begingroup$
Try writing it as $f(psi) = langle psi, grangle$ for some function $g$. Then find $|g|_2$.
$endgroup$
– Prahlad Vaidyanathan
May 20 '16 at 10:35
$begingroup$
Try writing it as $f(psi) = langle psi, grangle$ for some function $g$. Then find $|g|_2$.
$endgroup$
– Prahlad Vaidyanathan
May 20 '16 at 10:35
$begingroup$
always forget such representation.. thanks and solved ;)
$endgroup$
– Kenneth Ng
May 20 '16 at 10:51
$begingroup$
always forget such representation.. thanks and solved ;)
$endgroup$
– Kenneth Ng
May 20 '16 at 10:51
1
1
$begingroup$
@KennethNg: In this case you might want to add your solution as an answer for further reference.
$endgroup$
– gerw
May 20 '16 at 11:23
$begingroup$
@KennethNg: In this case you might want to add your solution as an answer for further reference.
$endgroup$
– gerw
May 20 '16 at 11:23
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
As pointed out by @KennethNg, using Riesz representation, and writing $f(psi)$ as $langlepsi,(3+isin(pi t)cos(pi t)rangle$, we get $$|f|=|3+isin(pi t)cos(pi t)|=int_0^13+frac{isin(2pi t)}{2}dt=3$$
answered Jan 26 at 17:29
vidyarthividyarthi
3,0611833
$endgroup$
add a comment |
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%2f1792810%2ffind-the-norm-of-a-linear-functional-in-l20-1%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
As pointed out by @KennethNg, using Riesz representation, and writing $f(psi)$ as $langlepsi,(3+isin(pi t)cos(pi t)rangle$, we get $$|f|=|3+isin(pi t)cos(pi t)|=int_0^13+frac{isin(2pi t)}{2}dt=3$$
answered Jan 26 at 17:29
vidyarthividyarthi
3,0611833
$endgroup$
add a comment |
$begingroup$
As pointed out by @KennethNg, using Riesz representation, and writing $f(psi)$ as $langlepsi,(3+isin(pi t)cos(pi t)rangle$, we get $$|f|=|3+isin(pi t)cos(pi t)|=int_0^13+frac{isin(2pi t)}{2}dt=3$$
answered Jan 26 at 17:29
vidyarthividyarthi
3,0611833
$endgroup$
add a comment |
$begingroup$
As pointed out by @KennethNg, using Riesz representation, and writing $f(psi)$ as $langlepsi,(3+isin(pi t)cos(pi t)rangle$, we get $$|f|=|3+isin(pi t)cos(pi t)|=int_0^13+frac{isin(2pi t)}{2}dt=3$$
answered Jan 26 at 17:29
vidyarthividyarthi
3,0611833
$endgroup$
As pointed out by @KennethNg, using Riesz representation, and writing $f(psi)$ as $langlepsi,(3+isin(pi t)cos(pi t)rangle$, we get $$|f|=|3+isin(pi t)cos(pi t)|=int_0^13+frac{isin(2pi t)}{2}dt=3$$
answered Jan 26 at 17:29
vidyarthividyarthi
3,0611833
answered Jan 26 at 17:29
vidyarthividyarthi
3,0611833
answered Jan 26 at 17:29
vidyarthividyarthi
3,0611833
answered Jan 26 at 17:29
vidyarthividyarthi
3,0611833
3,0611833
add a comment |
add a comment |
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%2f1792810%2ffind-the-norm-of-a-linear-functional-in-l20-1%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
1
$begingroup$
Try writing it as $f(psi) = langle psi, grangle$ for some function $g$. Then find $|g|_2$.
$endgroup$
– Prahlad Vaidyanathan
May 20 '16 at 10:35
$begingroup$
always forget such representation.. thanks and solved ;)
$endgroup$
– Kenneth Ng
May 20 '16 at 10:51
1
$begingroup$
@KennethNg: In this case you might want to add your solution as an answer for further reference.
$endgroup$
– gerw
May 20 '16 at 11:23