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I cannot understand this excerpt from a book on quantum mechanics with reference to observables.

I understand that the orthogonality theorem causes all values except the "equal" values lead to zero.However what do they mean when they say "if the integrand is finite the integral (29) vanishes, and if this holds for all ... <X|Y> vanishes..."

They then claim that in "general <X|Y> does not vanish...

Essentially i cannot garner where such assumptions are formed.Primarily why finite integrals causes the inner product to vanish as well as how it can just be assumed it normally does not vanish.

I understand that the orthogonality theorem causes all values except the "equal" values lead to zero.However what do they mean when they say "if the integrand is finite the integral (29) vanishes, and if this holds for all ... <X|Y> vanishes..."

They then claim that in "general <X|Y> does not vanish...

Essentially i cannot garner where such assumptions are formed.Primarily why finite integrals causes the inner product to vanish as well as how it can just be assumed it normally does not vanish.