c (IS) E (0,00) is continuous and non-increasing. iv) The norm 11·11 := 11·ll x + liB. liz is equivalent to the norm II Iv in v. 8) are satisfied. Then there exist constants c, 0: w (p,E) >0 such that for all p,E >0 with ~ ; ; ; 0 ; ; ; c Ey (~) • Proof: Let u E V with IIAUlly;;;;;p and IIBullz ; ; ; E.

4 are selfadjoint. 3 The singular value decomposition Let X and Y be nontrivial Hilbert spaces over the scalar field ~ . Throughout this section we don't distinguish the inner products and norms in X and Y. 11 Suppose T : X ~ Y is a linear mapping. Then the following conditions are equivalent: a) T is finite rank operator. b) There exist a number mEIN, an orthonormal set f l ' ... , fm in Y and linear independent vectors e 1 , ...

23) indicates if we choose the regularizing parameter in the way mentioned above. 1) again. In many cases one is not interested in the reconstruction of x O itself but only in a smeared object (u,xo)x where u is a given smearing element in X. For example this is usually the case in the reconstruction of objects from radiographs; the quantity (u,x o ) is then a local average of x O over some given resolving length. This means that we want to reconstruct the linear functional X :1 X ( u, 1-> x) E IR in x O from the data yE E Y using the information x E K : = {x E V I I Bxll ;;; E} and yEE RE(A) := {yE yl ~y -Axil;;; E for some x EX}.