The problems
Functional bases of numerical analysis

Let X and Y be two vector spaces and T : X → Y. Let x ∊ X and y ∊ Y. Consider the equation

There are three types of problems in numerical analysis:

• the direct problem: given T and x, calculate y (e.g. a definite integral);

• the inverse problem: given T and y, find x (e.g. systems of equations, differential and integral equations);

• the identification problem: given x and y, find T (e.g. function approximation).

When X and Y are of infinite dimension, a direct treatment of the problem is...