Formulation of the inverse problem in general mathematical spaces

1 Formulation of the inverse problem in general mathematical spaces 

In the first Chapter we have introduced an inverse problem as the solution of the operator equation 

This is exactly the moment when we have to introduce some kind of distance between two data sets that will help us to evaluate the accuracy of the inverse problem solution. In other words, we need to introduce a geometry to measure the distance between the actual and predicted data. The mathematical theory of function spaces provides us with guidance to the solution of this problem. The basic principles of this 

Let us assume that we are given two Banach (complete normed) spaces, M and D, and an operator A that acts from the space M to the space D  

We will call D a space of data sets and M a space of the model parameters. Operator A is a forward modeling operator that transforms any model m into the corresponding data d. The inverse problem is formulated as the solution of the operator equation (2.2). 

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