Definition of Bilinear functionals
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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 '18 at 10:34
Aakash GuptaAakash Gupta
333
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add a comment |
$begingroup$
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 '18 at 10:34
Aakash GuptaAakash Gupta
333
$endgroup$
add a comment |
$begingroup$
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 '18 at 10:34
Aakash GuptaAakash Gupta
333
$endgroup$
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 '18 at 10:34
Aakash GuptaAakash Gupta
333
asked Dec 6 '18 at 10:34
Aakash GuptaAakash Gupta
333
asked Dec 6 '18 at 10:34
Aakash GuptaAakash Gupta
333
asked Dec 6 '18 at 10:34
Aakash GuptaAakash Gupta
333
asked Dec 6 '18 at 10:34
Aakash GuptaAakash Gupta
333
333
add a comment |
add a comment |
1 Answer
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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 '18 at 10:43
littleOlittleO
29.8k646109
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why is $phi(x,y)$ = $eta_x(y)$ = $xi_y(x)$? What does that signify?
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– Aakash Gupta
Dec 6 '18 at 10:47
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@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.
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– gandalf61
Dec 6 '18 at 11:01
add a comment |
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1 Answer
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1 Answer
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$begingroup$
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 '18 at 10:43
littleOlittleO
29.8k646109
$endgroup$
$begingroup$
why is $phi(x,y)$ = $eta_x(y)$ = $xi_y(x)$? What does that signify?
$endgroup$
– Aakash Gupta
Dec 6 '18 at 10:47
$begingroup$
@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.
$endgroup$
– gandalf61
Dec 6 '18 at 11:01
add a comment |
$begingroup$
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 '18 at 10:43
littleOlittleO
29.8k646109
$endgroup$
$begingroup$
why is $phi(x,y)$ = $eta_x(y)$ = $xi_y(x)$? What does that signify?
$endgroup$
– Aakash Gupta
Dec 6 '18 at 10:47
$begingroup$
@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.
$endgroup$
– gandalf61
Dec 6 '18 at 11:01
add a comment |
$begingroup$
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 '18 at 10:43
littleOlittleO
29.8k646109
$endgroup$
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 '18 at 10:43
littleOlittleO
29.8k646109
edited Dec 6 '18 at 10:44
answered Dec 6 '18 at 10:43
littleOlittleO
29.8k646109
answered Dec 6 '18 at 10:43
littleOlittleO
29.8k646109
answered Dec 6 '18 at 10:43
littleOlittleO
29.8k646109
29.8k646109
$begingroup$
why is $phi(x,y)$ = $eta_x(y)$ = $xi_y(x)$? What does that signify?
$endgroup$
– Aakash Gupta
Dec 6 '18 at 10:47
$begingroup$
@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.
$endgroup$
– gandalf61
Dec 6 '18 at 11:01
add a comment |
$begingroup$
why is $phi(x,y)$ = $eta_x(y)$ = $xi_y(x)$? What does that signify?
$endgroup$
– Aakash Gupta
Dec 6 '18 at 10:47
$begingroup$
@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.
$endgroup$
– gandalf61
Dec 6 '18 at 11:01
$begingroup$
why is $phi(x,y)$ = $eta_x(y)$ = $xi_y(x)$? What does that signify?
$endgroup$
– Aakash Gupta
Dec 6 '18 at 10:47
$begingroup$
why is $phi(x,y)$ = $eta_x(y)$ = $xi_y(x)$? What does that signify?
$endgroup$
– Aakash Gupta
Dec 6 '18 at 10:47
$begingroup$
@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.
$endgroup$
– gandalf61
Dec 6 '18 at 11:01
$begingroup$
@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.
$endgroup$
– gandalf61
Dec 6 '18 at 11:01
add a comment |
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