A mapping of one set into another, each of which has a certain structure (defined by algebraic operations, a topology, or by an order relation). The general definition of an operator coincides with the definition of a mapping or function. Let $X$ and $Y$ be two sets.
Using C++ standardese, the function call syntax (operator+(x, y) or x.operator+(y)) works only for operator functions: 13.5 Overloaded operators [over.oper]. 4. Operator functions are usually not called directly; instead they are invoked to evaluate the operators they implement (13.5.1 - 13.5.7).
A mapping in which the law of the correspondence is given by an integral. An integral operator is sometimes called an integral transformation. Thus, for Urysohn's integral operator (cf. Urysohn equation) , the law of the correspondence is given by the integral...
linear transformation, linear map. A mapping between two vector spaces (cf. Vector space) that is compatible with their linear structures. More precisely, a mapping , where and are vector spaces over a field , is called a linear operator from to if. for all , .