Understanding the Quadratic Form Identity in Two-Variable Equations

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Discussion Overview

The discussion revolves around the concept of a quadratic form identity in the context of two-variable equations, particularly focusing on an equality referenced as (4.26). Participants explore the implications of boundedness in L2 spaces and the continuity of a function w, questioning how these factors relate to the identity operator on L2(X).

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant asks for clarification on why a specific equality is considered a quadratic form identity and its relation to two variables.
  • Another participant questions the meaning of boundedness in L2 and its connection to the quadratic form identity on S.
  • There is inquiry into whether the continuity of w implies that the two terms yield the same result for every vector, along with a request for proof of this assertion.
  • Several participants discuss issues with accessing a linked resource, with varying success reported.
  • A participant seeks clarification on the meaning of the symbol ##\chi##.
  • Another participant suggests that ##\chi## may be identified with R`^d.
  • There is a question regarding the association of complex numbers to pairs of functions, referencing Dirac notation and its implications in the context of the discussion.

Areas of Agreement / Disagreement

The discussion contains multiple competing views and questions, with no consensus reached on the interpretations of the quadratic form identity or the implications of continuity and boundedness.

Contextual Notes

Participants express uncertainty regarding the definitions and implications of terms used, such as "quadratic form identity," "boundedness," and "identity operator," indicating a need for further clarification on these concepts.

Who May Find This Useful

This discussion may be of interest to those studying functional analysis, particularly in relation to quadratic forms, L2 spaces, and operator theory.

Heidi
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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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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.
 
is there a problem for everybody? (it works for me).
 
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
 
What is ##\chi##?
 
it may be identified with R`^d
 
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> ?
 

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