Definition of metric space

The answer is very simple. In the first chapter of this text we have formulated 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, which will help us to evaluate the accuracy of the inverse problem solution. That is why we need to introduce a geometry to measure the distance between actual and predicted data. The mathematical theory of function spaces provides us with guidance in the solution of this problem. The simplest and, at the same time, the most important space which contains a geometry (in the sense that there is a distance between any two elements of this space) is the metric space. 

A metric space is a set M of elements h for each two of which the nonnegative number /x(h, g) can be determined, called the distance between the two elements h and g or metric. The metric has to satisfy the following conditions  

One important property of the metric space is that we can introduce an idea of convergence of a sequence of elements in this space. 

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