Prehilbert spaces

Many problems admit a variational interpretation, i.e. a formulation in which a solution to the problem at hand achieves an extremum for a certain functional, such as energy. This point of view originated in physics, but is frequently encountered in other sciences, notably mathematics. On the other hand, a problem posed in the form of an equation ϕ(x) = 0 may not admit an exact solution, and the search for approximate solutions again leads to an interest in objects that minimize ϕ, when this function does not cancel out: this is the case with the method of least squares.

The search for extremums of a function has many facets, but the two examples we have given refer to a framework in which the distance is quadratic in nature. In the case of the least-squares method, a Euclidean norm is minimized. The framework is therefore that of finite-dimensional spaces, where the notion of orthogonal projection onto a finite-dimensional vector subspace is easy to understand and detail. In the case of energy minimization, we are led to minimize a functional of the type , where the argument is a real or complex function f, whose argument is a spatial or temporal variable, this function describing a certain set D defined by the constraints of the problem. By integrating over a spatial or temporal domain, we define a quantity which, when f describes D, must be minimized.

Such a functional actually derives from a real or complex pre-Hilbertian scalar product, and we can see that finding the minimum of on D means finding the distance from 0 to D for the associated pre-Hilbertian norm, as well...