Definition of Bilinear functionals
I am reading the book 'Introduction to hilbert space and the theory of spectral multiplicity' and in chapter 1, section 2, it gives the definition of bilinear functional as
A bilinear functional on a complex vector space $C$ is a complex valued function $phi$ on the cartesian product of $C$ with itself such that if $xi_y(x) = eta_x(y) = phi(x,y)$, then for every $x$ and $y$ in $C$, $xi_y$ is a linear functional and $eta_x$ is a conjugate linear functional.
I tried to search through the internet, but couldn't find the such a definition of Bilinear functionals. Could someone please help me understand it?
PS: I don't have a formal math background but I learnt a lot of math in my engineering curriculum.
linear-algebra
asked Dec 6 at 10:34
Aakash Gupta
333
add a comment |
I am reading the book 'Introduction to hilbert space and the theory of spectral multiplicity' and in chapter 1, section 2, it gives the definition of bilinear functional as
A bilinear functional on a complex vector space $C$ is a complex valued function $phi$ on the cartesian product of $C$ with itself such that if $xi_y(x) = eta_x(y) = phi(x,y)$, then for every $x$ and $y$ in $C$, $xi_y$ is a linear functional and $eta_x$ is a conjugate linear functional.
I tried to search through the internet, but couldn't find the such a definition of Bilinear functionals. Could someone please help me understand it?
PS: I don't have a formal math background but I learnt a lot of math in my engineering curriculum.
linear-algebra
asked Dec 6 at 10:34
Aakash Gupta
333
add a comment |
I am reading the book 'Introduction to hilbert space and the theory of spectral multiplicity' and in chapter 1, section 2, it gives the definition of bilinear functional as
A bilinear functional on a complex vector space $C$ is a complex valued function $phi$ on the cartesian product of $C$ with itself such that if $xi_y(x) = eta_x(y) = phi(x,y)$, then for every $x$ and $y$ in $C$, $xi_y$ is a linear functional and $eta_x$ is a conjugate linear functional.
I tried to search through the internet, but couldn't find the such a definition of Bilinear functionals. Could someone please help me understand it?
PS: I don't have a formal math background but I learnt a lot of math in my engineering curriculum.
linear-algebra
asked Dec 6 at 10:34
Aakash Gupta
333
I am reading the book 'Introduction to hilbert space and the theory of spectral multiplicity' and in chapter 1, section 2, it gives the definition of bilinear functional as
A bilinear functional on a complex vector space $C$ is a complex valued function $phi$ on the cartesian product of $C$ with itself such that if $xi_y(x) = eta_x(y) = phi(x,y)$, then for every $x$ and $y$ in $C$, $xi_y$ is a linear functional and $eta_x$ is a conjugate linear functional.
I tried to search through the internet, but couldn't find the such a definition of Bilinear functionals. Could someone please help me understand it?
PS: I don't have a formal math background but I learnt a lot of math in my engineering curriculum.
linear-algebra
linear-algebra
asked Dec 6 at 10:34
Aakash Gupta
333
asked Dec 6 at 10:34
Aakash Gupta
333
asked Dec 6 at 10:34
Aakash Gupta
333
asked Dec 6 at 10:34
Aakash Gupta
333
asked Dec 6 at 10:34
Aakash Gupta
333
333
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
Maybe this is more clear: To say that a function $phi:C times C to mathbb C$ is bilinear means that if $x, y in C$ then the function $phi(cdot, y)$ is linear and the function $phi(x, cdot)$ is conjugate linear.
In the passage you quoted, the function $phi(cdot,y)$ is denoted $xi_y$ and the function $phi(x,cdot)$ is denoted $eta_x$.
answered Dec 6 at 10:43
littleO
28.4k642104
why is $phi(x,y)$ = $eta_x(y)$ = $xi_y(x)$? What does that signify?
– Aakash Gupta
Dec 6 at 10:47
@AakashGupta This simply means that we can think of $phi(x,y)$ as a function of $x$ and $y$ or as a family of functions of $y$, parameterised by $x$ or as a family of functions of $x$, parameterised by $y$. If you think of $phi(x,y)$ as a surface then $eta_x(y)$ and $xi_y(x)$ are like cross- sections of that surface.
– gandalf61
Dec 6 at 11:01
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%2f3028324%2fdefinition-of-bilinear-functionals%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
Maybe this is more clear: To say that a function $phi:C times C to mathbb C$ is bilinear means that if $x, y in C$ then the function $phi(cdot, y)$ is linear and the function $phi(x, cdot)$ is conjugate linear.
In the passage you quoted, the function $phi(cdot,y)$ is denoted $xi_y$ and the function $phi(x,cdot)$ is denoted $eta_x$.
answered Dec 6 at 10:43
littleO
28.4k642104
why is $phi(x,y)$ = $eta_x(y)$ = $xi_y(x)$? What does that signify?
– Aakash Gupta
Dec 6 at 10:47
@AakashGupta This simply means that we can think of $phi(x,y)$ as a function of $x$ and $y$ or as a family of functions of $y$, parameterised by $x$ or as a family of functions of $x$, parameterised by $y$. If you think of $phi(x,y)$ as a surface then $eta_x(y)$ and $xi_y(x)$ are like cross- sections of that surface.
– gandalf61
Dec 6 at 11:01
add a comment |
Maybe this is more clear: To say that a function $phi:C times C to mathbb C$ is bilinear means that if $x, y in C$ then the function $phi(cdot, y)$ is linear and the function $phi(x, cdot)$ is conjugate linear.
In the passage you quoted, the function $phi(cdot,y)$ is denoted $xi_y$ and the function $phi(x,cdot)$ is denoted $eta_x$.
answered Dec 6 at 10:43
littleO
28.4k642104
why is $phi(x,y)$ = $eta_x(y)$ = $xi_y(x)$? What does that signify?
– Aakash Gupta
Dec 6 at 10:47
@AakashGupta This simply means that we can think of $phi(x,y)$ as a function of $x$ and $y$ or as a family of functions of $y$, parameterised by $x$ or as a family of functions of $x$, parameterised by $y$. If you think of $phi(x,y)$ as a surface then $eta_x(y)$ and $xi_y(x)$ are like cross- sections of that surface.
– gandalf61
Dec 6 at 11:01
add a comment |
Maybe this is more clear: To say that a function $phi:C times C to mathbb C$ is bilinear means that if $x, y in C$ then the function $phi(cdot, y)$ is linear and the function $phi(x, cdot)$ is conjugate linear.
In the passage you quoted, the function $phi(cdot,y)$ is denoted $xi_y$ and the function $phi(x,cdot)$ is denoted $eta_x$.
answered Dec 6 at 10:43
littleO
28.4k642104
Maybe this is more clear: To say that a function $phi:C times C to mathbb C$ is bilinear means that if $x, y in C$ then the function $phi(cdot, y)$ is linear and the function $phi(x, cdot)$ is conjugate linear.
In the passage you quoted, the function $phi(cdot,y)$ is denoted $xi_y$ and the function $phi(x,cdot)$ is denoted $eta_x$.
answered Dec 6 at 10:43
littleO
28.4k642104
edited Dec 6 at 10:44
answered Dec 6 at 10:43
littleO
28.4k642104
answered Dec 6 at 10:43
littleO
28.4k642104
answered Dec 6 at 10:43
littleO
28.4k642104
28.4k642104
why is $phi(x,y)$ = $eta_x(y)$ = $xi_y(x)$? What does that signify?
– Aakash Gupta
Dec 6 at 10:47
@AakashGupta This simply means that we can think of $phi(x,y)$ as a function of $x$ and $y$ or as a family of functions of $y$, parameterised by $x$ or as a family of functions of $x$, parameterised by $y$. If you think of $phi(x,y)$ as a surface then $eta_x(y)$ and $xi_y(x)$ are like cross- sections of that surface.
– gandalf61
Dec 6 at 11:01
add a comment |
why is $phi(x,y)$ = $eta_x(y)$ = $xi_y(x)$? What does that signify?
– Aakash Gupta
Dec 6 at 10:47
@AakashGupta This simply means that we can think of $phi(x,y)$ as a function of $x$ and $y$ or as a family of functions of $y$, parameterised by $x$ or as a family of functions of $x$, parameterised by $y$. If you think of $phi(x,y)$ as a surface then $eta_x(y)$ and $xi_y(x)$ are like cross- sections of that surface.
– gandalf61
Dec 6 at 11:01
why is $phi(x,y)$ = $eta_x(y)$ = $xi_y(x)$? What does that signify?
– Aakash Gupta
Dec 6 at 10:47
why is $phi(x,y)$ = $eta_x(y)$ = $xi_y(x)$? What does that signify?
– Aakash Gupta
Dec 6 at 10:47
@AakashGupta This simply means that we can think of $phi(x,y)$ as a function of $x$ and $y$ or as a family of functions of $y$, parameterised by $x$ or as a family of functions of $x$, parameterised by $y$. If you think of $phi(x,y)$ as a surface then $eta_x(y)$ and $xi_y(x)$ are like cross- sections of that surface.
– gandalf61
Dec 6 at 11:01
@AakashGupta This simply means that we can think of $phi(x,y)$ as a function of $x$ and $y$ or as a family of functions of $y$, parameterised by $x$ or as a family of functions of $x$, parameterised by $y$. If you think of $phi(x,y)$ as a surface then $eta_x(y)$ and $xi_y(x)$ are like cross- sections of that surface.
– gandalf61
Dec 6 at 11:01
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.
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%2f3028324%2fdefinition-of-bilinear-functionals%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
