B Understanding the Quadratic Form Identity in Two-Variable Equations

[Heidi]

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.

 

[WWGD]

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.

 

[Heidi]

is there a problem for everybody? (it works for me).

 

[fresh_42]

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

 

[WWGD]

What is $\chi$?

 

[Heidi]

it may be identified with R`^d

 

[Heidi]

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> ?