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M are arbitrary complex numbers and ϕ denotes a particular solution of the inhomogeneous equation. 4) has only the trivial solution ϕ = 0. 5 for integral equations of the second kind with continuous or weakly singular kernels. 6. , S : X → Y is an isomorphism, and A : X → Y is a compact linear operator from a normed space X into a normed space Y. Proof. 21. 3 generates an operator P : X → N(Lr ) that maps ϕ ∈ X onto Pϕ := ψ defined by the unique decomposition ϕ = ψ + χ with ψ ∈ N(Lr ) and χ ∈ Lr (X).

N ) is a normed space with the maximum norm ϕ ∞ := max ϕi . ,n Let Aik : Xk → Xi , i, k = 1, . . , n, be linear operators. Show that the matrix operator A : X → X defined by n Aik ϕk (Aϕ)i := k=1 is bounded or compact if and only if each of its components Aik : Xk → Xi is bounded or compact, respectively. 4 for systems of operator and integral equations of the second kind. 5. Show that the integral operator with continuous kernel ∞ K(x, y) := k=0 1 {cos(k + 1)x sin ky − sin(k + 1)x cos ky} (k + 1)2 on the interval [0, 2π] has no eigenvalues.

10) by ds(y) ≤ 2ρm−2 dρdω, and that the projection of S [x; R] onto the tangent plane is contained in the disk of radius R and center x. Furthermore, we have ∂D\S [x;R] K(x, y)ϕ(y) ds(y) ≤ M ϕ ≤M ϕ ∞ ∞R Rα−m+1 ds(y) ∂D\S [x;R] α−m+1 |∂D|. 8) exists as an improper integral. 29. 1. Let A : X → Y be a bounded linear operator from a normed space X into a normed space Y and let X and Y be the completions of X and Y, respectively. Then there exists a uniquely determined bounded linear operator A : X → Y such that Aϕ = Aϕ for all ϕ ∈ X.