By Vandebril R., Van Barel M., Golub G.

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The nullspace of the bounded linear operator L is a closed subspace of X, since for each sequence (ϕn ) with ϕn → ϕ, n → ∞, and Lϕn = 0 we have that Lϕ = 0. Each ϕ ∈ N(L) satisfies Aϕ = ϕ, and therefore the restriction of A to N(L) coincides with the identity operator on N(L). The operator A is compact on X and therefore also compact from N(L) into N(L), since N(L) is closed. 25. 2 (Second Riesz Theorem). , L(X) := {Lϕ : ϕ ∈ X}, is a closed linear subspace. Proof. The range of the linear operator L is a linear subspace.

29 we can prove the following theorem. 30. Integral operators with continuous or weakly singular kernel are compact linear operators on C(∂D) if ∂D is of class C 1 . Proof. 27 essentially remains unaltered. 8). Since the surface ∂D is of class C 1 , the normal vector ν is continuous on ∂D. 10) for all x, y ∈ ∂D with |x−y| ≤ R. Furthermore, we can assume that R is small enough such that the set S [x; R] := {y ∈ ∂D : |y − x| ≤ R} is connected for each x ∈ ∂D. 10) implies that S [x; R] can be projected bijectively onto the tangent plane to ∂D at the point x.

In particular, this implies that the boundary ∂D can be represented locally by a parametric representation x(u) = (x1 (u), . . , xm (u)) mapping an open parameter domain U ⊂ IRm−1 bijectively onto a surface patch S of ∂D with the property that the vectors ∂x , ∂ui i = 1, . . , m − 1, are linearly independent at each point x of S . Such a parameterization we call a regular parametric representation. The whole boundary ∂D is obtained by matching a finite number of such surface patches. On occasion, we will express the property of a domain D to be of class C n also by saying that its boundary ∂D is of class C n .

### A bibliography on semiseparable matrices by Vandebril R., Van Barel M., Golub G.

by Richard

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