Verifying an inner product
$begingroup$
Let $E$ be the linear space of all continuous complex-valued functions defined on the interval $(a,b)$ of the real line. Consider the inner product in $E$
$langle f,g rangle = int_a^{b} f(x) overline {g(x)} dx$
Answer
Linearity in the first argument is easy
Now,
$langle f,f rangle = int_a^b f(x) overline{f(x)} dx = int_a^b f(x)^2 dx geq 0$ by Chebyshev inequality.
let $f(x)=0$ then $int_a^b 0 dx =0= langle f,f rangle =0$
let $langle f,f rangle =0 implies int_a^b f(x)^2 dx =0 implies f(x)=0$ a.e on $(a,b)$ thus
$langle f(x),f(x) rangle=0 iff f(x)=0$.
For the conjugate symmetry,
$langle f,g rangle = int_a^b f overline g dx = int_a^b overline{overline f} overline g dx = int_a^b overline g overline{overline f} dx = int_a^b overline{g overline{f}} dx$=$overline{langle g,f rangle}$
functional-analysis proof-verification proof-writing inner-product-space
asked Dec 25 '18 at 6:38
Dreamer123Dreamer123
32329
$endgroup$
add a comment |
$begingroup$
Let $E$ be the linear space of all continuous complex-valued functions defined on the interval $(a,b)$ of the real line. Consider the inner product in $E$
$langle f,g rangle = int_a^{b} f(x) overline {g(x)} dx$
Answer
Linearity in the first argument is easy
Now,
$langle f,f rangle = int_a^b f(x) overline{f(x)} dx = int_a^b f(x)^2 dx geq 0$ by Chebyshev inequality.
let $f(x)=0$ then $int_a^b 0 dx =0= langle f,f rangle =0$
let $langle f,f rangle =0 implies int_a^b f(x)^2 dx =0 implies f(x)=0$ a.e on $(a,b)$ thus
$langle f(x),f(x) rangle=0 iff f(x)=0$.
For the conjugate symmetry,
$langle f,g rangle = int_a^b f overline g dx = int_a^b overline{overline f} overline g dx = int_a^b overline g overline{overline f} dx = int_a^b overline{g overline{f}} dx$=$overline{langle g,f rangle}$
functional-analysis proof-verification proof-writing inner-product-space
asked Dec 25 '18 at 6:38
Dreamer123Dreamer123
32329
$endgroup$
add a comment |
$begingroup$
Let $E$ be the linear space of all continuous complex-valued functions defined on the interval $(a,b)$ of the real line. Consider the inner product in $E$
$langle f,g rangle = int_a^{b} f(x) overline {g(x)} dx$
Answer
Linearity in the first argument is easy
Now,
$langle f,f rangle = int_a^b f(x) overline{f(x)} dx = int_a^b f(x)^2 dx geq 0$ by Chebyshev inequality.
let $f(x)=0$ then $int_a^b 0 dx =0= langle f,f rangle =0$
let $langle f,f rangle =0 implies int_a^b f(x)^2 dx =0 implies f(x)=0$ a.e on $(a,b)$ thus
$langle f(x),f(x) rangle=0 iff f(x)=0$.
For the conjugate symmetry,
$langle f,g rangle = int_a^b f overline g dx = int_a^b overline{overline f} overline g dx = int_a^b overline g overline{overline f} dx = int_a^b overline{g overline{f}} dx$=$overline{langle g,f rangle}$
functional-analysis proof-verification proof-writing inner-product-space
asked Dec 25 '18 at 6:38
Dreamer123Dreamer123
32329
$endgroup$
Let $E$ be the linear space of all continuous complex-valued functions defined on the interval $(a,b)$ of the real line. Consider the inner product in $E$
$langle f,g rangle = int_a^{b} f(x) overline {g(x)} dx$
Answer
Linearity in the first argument is easy
Now,
$langle f,f rangle = int_a^b f(x) overline{f(x)} dx = int_a^b f(x)^2 dx geq 0$ by Chebyshev inequality.
let $f(x)=0$ then $int_a^b 0 dx =0= langle f,f rangle =0$
let $langle f,f rangle =0 implies int_a^b f(x)^2 dx =0 implies f(x)=0$ a.e on $(a,b)$ thus
$langle f(x),f(x) rangle=0 iff f(x)=0$.
For the conjugate symmetry,
$langle f,g rangle = int_a^b f overline g dx = int_a^b overline{overline f} overline g dx = int_a^b overline g overline{overline f} dx = int_a^b overline{g overline{f}} dx$=$overline{langle g,f rangle}$
functional-analysis proof-verification proof-writing inner-product-space
functional-analysis proof-verification proof-writing inner-product-space
asked Dec 25 '18 at 6:38
Dreamer123Dreamer123
32329
asked Dec 25 '18 at 6:38
Dreamer123Dreamer123
32329
asked Dec 25 '18 at 6:38
Dreamer123Dreamer123
32329
asked Dec 25 '18 at 6:38
Dreamer123Dreamer123
32329
asked Dec 25 '18 at 6:38
Dreamer123Dreamer123
32329
32329
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The only mistake is in writing $zoverline {z}=z^{2}$ and $z^{2}geq 0$. For a complex number $z$ we have $zoverline {z}=|z|^{2} geq0$. ($z^{2} geq 0$ is not true but $|z|^{2} geq 0)$.
answered Dec 25 '18 at 6:43
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
$endgroup$
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%2f3051895%2fverifying-an-inner-product%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
$begingroup$
The only mistake is in writing $zoverline {z}=z^{2}$ and $z^{2}geq 0$. For a complex number $z$ we have $zoverline {z}=|z|^{2} geq0$. ($z^{2} geq 0$ is not true but $|z|^{2} geq 0)$.
answered Dec 25 '18 at 6:43
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
$endgroup$
add a comment |
$begingroup$
The only mistake is in writing $zoverline {z}=z^{2}$ and $z^{2}geq 0$. For a complex number $z$ we have $zoverline {z}=|z|^{2} geq0$. ($z^{2} geq 0$ is not true but $|z|^{2} geq 0)$.
answered Dec 25 '18 at 6:43
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
$endgroup$
add a comment |
$begingroup$
The only mistake is in writing $zoverline {z}=z^{2}$ and $z^{2}geq 0$. For a complex number $z$ we have $zoverline {z}=|z|^{2} geq0$. ($z^{2} geq 0$ is not true but $|z|^{2} geq 0)$.
answered Dec 25 '18 at 6:43
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
$endgroup$
The only mistake is in writing $zoverline {z}=z^{2}$ and $z^{2}geq 0$. For a complex number $z$ we have $zoverline {z}=|z|^{2} geq0$. ($z^{2} geq 0$ is not true but $|z|^{2} geq 0)$.
answered Dec 25 '18 at 6:43
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
edited Dec 25 '18 at 7:30
answered Dec 25 '18 at 6:43
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
answered Dec 25 '18 at 6:43
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
answered Dec 25 '18 at 6:43
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
59.2k42161
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%2f3051895%2fverifying-an-inner-product%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
