Integral Equations

is called the Euclidian space.In a complex Euclidian space the inner product satisfies(x, y) = (y, x),(x 1 + x 2 , y) = (x 1 , y) + (x 2 , y),(λx, y) = λ(x, y),Note that in a complex linear space(x, x) ≥ 0, (x, x) = 0 if and only if x = 0.(x, λy) = ¯λ(x, y).The norm in the Euclidian space is introduced by||x| = √ x, x.The Hilbert space H is a complete (in the metric ρ(x, y) = ||x − y||) infinite-dimensionalEuclidian space.In the Hilbert space H, the operator A ∗ adjoint to an operator A is defined by(Ax, y) = (x, A ∗ y), ∀x, y ∈ H.The selfadjoint operator A = A ∗ is defined from(Ax, y) = (x, Ay), ∀x, y ∈ H.10.2 Completely continuous integral operators in Hilbert spaceConsider an IE of the second kindφ(x) = f(x) +∫ baK(x, y)φ(y)dy. (179)Assume that K(x, y) is a Hilbert–Schmidt kernel, i.e., a square-integrable function in the squareΠ = {(x, y) : a ≤ x ≤ b, a ≤ y ≤ b}, so thatand f(x) ∈ L 2 [a, b], i.e.,∫ b ∫ baa∫ ba|K(x, y)| 2 dxdy ≤ ∞, (180)|f(x)| 2 dx ≤ ∞.Define a linear Fredholm (integral) operator corresponding to IE (179)Aφ(x) =∫ baK(x, y)φ(y)dy. (181)If K(x, y) is a Hilbert–Schmidt kernel, then operator (181) will be called a Hilbert–Schmidtoperator.Rewrite IE (179) as a linear operator equationφ = Aφ(x) + f, f, φ ∈ L 2 [a, b]. (182)35