6.12 (Weighted L²-spaces) Let r : R →]0, [infinity][ be a continuous function, and define the vector space L2r (R) by
L2r (R) := {ƒ : R¬C | |*~_ \f(x)^2r(x) dx < [infinity]}.

(i) Show that (f₁9) L2r (R) := [infinity]J[infinity] f(x)g(x)r(x) dx (6.23) defines an inner product on L2r (R).
(ii) Using that L²(R) is a Hilbert space, show that L2r (R) equipped with the inner product in (6.23) is a Hilbert space.