Basic Definitions and Notations

We are dealing with a molecular system consisting of N atoms which follows the Born-Oppenheimer approximation. Such a system possesses in general a well-defined structure and can be completely described by the Cartesian coordinates 

When we choose the n-dimensional Euclidian space E (cf. Ref.2), which is associated with the vector space R, as the configuration space, then the possible nuclear arrangements of a molecular system may be identified with the points of E, and the n-tupels of Eq. (1) are position vectors that describe the points of E with respect to the chosen Cartesian coordinate system. The forces acting on the nuclei of the system may be identified with the vectors of r . So we have always to distinguish between position vectors that define a point of E (we shall call it points) and vectors that describe a force or a displacement. 

The vector space R may be equipped with the inner product x y  

Sometimes it is useful to restrict the freedom of movement of some nuclei of a molecular system. In such a situation the possible arrangements of the nuclei correspond to points of a Euclidian space E with n 3N. Therefore, in the following the number n indicates an arbitrary dimension, which is chosen in accordance with the problem under consideration. 

Since not all points of E correspond to chemically meaningful arrangements, we shall confine always to domains 2) of R which do not contain vectors that match pathological nuclear arrangements. 

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In this appendix we will discuss the mathematical tools necessary for developing the regularization theory of inverse problem solutions. They are based on the methods of functional analysis, which employ the ideas of functional spaces. Thus we should start our discussion by introducing the basic definitions and notations from functional analysis. Before doing so, I remind the reader of the basic properties of the simplest and, at the same time, the most fundamental mathematical space - Euclidean space. 

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