Operators in functional spaces

We can treat the elements of functional spaces as geometrical points, and we can consider different transformations of these points from one into another. These transformations can be described by corresponding rules, which are called operators. We give now a more strict definition of an operator. 

Definition 53 Let X and Y be metric spaces and D some subdomain of X D C X. If for any x G D, we can assign according to a certain rule some element y G Y we say that the operator A is given on D with the values in Y  

we can see that the operator is a natural generalization of the function for an abstract mathematical space. 

We know that one of the very important properties of a function is whether it is continuous or discontinuous. This property can be applied to operators as well. 

Definition 54 Let X and Y be metric spaces with metrics (x, x ) and P2 (y y ) and let A be an operator, transforming elements of X into Y. A is said to be continuous at a point xq in X if for each real number e 0 there exists a real number (5 0 such that, for any two elements x, Xg G X, satisfying the condition Xg) 6, the distance between the results of their transformation by an A operator is less than e ( ) ( o))  

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If n ranges from — oo to + oo it is possible to regard E as an actual operator in function space, but if there are one or two boundaries it is better to regard E simply as an abbreviating symbol. Most of its properties are immediately evident, but we note in particular... 

The theory also explains the value of 5. Unfortunately, that part of the story requires more sophisticated apparatus than we are prepared to discuss (operators in function space, Frechet derivatives, etc.). Instead we turn now to a concrete example of renormalization. The calculations are only approximate, but they can be done explicitly, using algebra instead of functional equations. 

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