Understanding the Quadratic Form Identity in Two-Variable Equations

In summary: The equality mentioned (4.26) involves a function w that depends on two variables. It is stated that if B is bounded (L2), then this equality becomes a quadratic form identity on S. This means that the function w satisfies certain conditions and can be expressed as a quadratic form. The author then explains that if w is continuous in one of the variables, it results in an identity operator on L2(X), meaning that for every vector, the two terms will give the same result. The author does not provide a proof for this and it is unclear how to go about proving it.
  • #1
Heidi
411
40
TL;DR Summary
could you explain why this equality is a quadratic form identity?
Summary: could you explain why this equality is a quadratic form identity?

i read this equality (4.26) here w depends on two variables. it is written that if B is bounded (L2) then it is a quadratic form identity on S. what does it mean? is it related to the two variables?
next the author writes that if w is continuous in the variable we have an identity operator on L2(X). does il mean that for every vector the two terms give a same result? how to prove that?

thanks a lot.
 
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  • #2
Heidi said:
Summary: could you explain why this equality is a quadratic form identity?

i read this equality (4.26) here w depends on two variables. it is written that if B is bounded (L2) then it is a quadratic form identity on S. what does it mean? is it related to the two variables?
next the author writes that if w is continuous in the variable we have an identity operator on L2(X). does il mean that for every vector the two terms give a same result? how to prove that?

thanks a lot.
Link does not seem to be working. Please use a screenshot or something else.
 
  • #3
is there a problem for everybody? (it works for me).
 
  • #4
Didn't work here either for the first time. The second time worked. Don't ask me why. FF-effect maybe.
https://books.google.fr/books?id=uZdNtduC5NAC&pg=PA103#v=onepage&q&f=false
1567807416884.png

1567807189576.png


1567807336013.png
 
  • #5
What is ##\chi##?
 
  • #6
it may be identified with R`^d
 
  • #7
Does it mean in the first case that we have a same way to associate a complex number to each couple f1, f2 of function? the dirac notation would be <f1|B|f2>
and in the second case the same function noted B|f> ?
 

1. What is a quadratic form identity?

A quadratic form identity is a mathematical expression that represents a quadratic function. It is written in the form of ax^2 + bx + c, where a, b, and c are constants and x is the variable. It is also known as a quadratic equation or a second-degree polynomial.

2. How is a quadratic form identity different from a linear form identity?

A quadratic form identity involves a squared term (ax^2) whereas a linear form identity does not. This means that a quadratic form identity will have a curved graph, while a linear form identity will have a straight line graph.

3. What is the importance of quadratic form identities in mathematics?

Quadratic form identities are important in mathematics because they have many real-world applications, such as in physics, engineering, and economics. They also help in solving complex mathematical problems and understanding the behavior of quadratic functions.

4. How do you solve a quadratic form identity?

To solve a quadratic form identity, you can use various methods such as factoring, completing the square, or using the quadratic formula. These methods involve manipulating the equation to isolate the variable x and find its value.

5. Can a quadratic form identity have more than one solution?

Yes, a quadratic form identity can have two solutions, also known as roots. These solutions can be real or complex numbers and can be found by solving the equation using the methods mentioned above. However, there are cases where a quadratic form identity may have no real solutions.

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