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 ${\displaystyle \int {\left|f\right|}^2}$ , 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 ${\left|f\right|}^2$ 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 ${\displaystyle \int {\left|f\right|}^2}$ on D means finding the distance from 0 to D for the associated pre-Hilbertian norm, as well as the f's in D that realize this minimum. If we stay within the framework of starting functions (e.g., continuous functions), there may be no solution, which simply means that there is no minimum, only an infimum. However, we find that such a problem always admits a unique solution when D is a closed vector subspace of a complete pre-Hilbert space, which we call a Hilbert space. There remains the difficulty of finding such a Hilbert space and interpreting it. It is not our purpose to solve it here,...