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I have a scalar function in a Hilbert space of the form f(x) = F(x)* · h o g(x) where F and h are vectors and I need to write < f, f >_L^2 in the shape ∫h(x) ∫H(x,y) h(y) dy dx.

My derivation is the following using delta function:

< f, f >_L^2 = ∫F(x)* · h o g(x) ∫δ(x-y) F(y)* h o g(x) dy dx

applying change of variables on both spatial integrals

< f, f >_L^2 = ∫|Dg-1(x)| F(g-1(x))* · h(x) ∫δ(g-1(x)-g-1(y)) |Dg-1(x)|F(g-1(y))* h(y) dy dx =

∫ h(x) ∫|Dg-1(x)| F(g-1(x))* δ(g-1(x)-g-1(y)) |Dg-1(x)|F(g-1(y))* h(y) dy dx

yielding

H(x,y) = |Dg-1(x)| F(g-1(x)) δ(g-1(x)-g-1(y)) |Dg-1(x)|F(g-1(y))*

¿Is the use of delta function somehow misleading?

¿In which situations (continuity, differentiability) may I

use this derivation?

¿Is my derivation correct?

Thank you so much.

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# Change of variables and use of dirac delta function

Can you offer guidance or do you also need help?

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