17.  From Geometry to Functional Analysis

The study of Euclidean sets is traditionally and naturally carried out using the concepts and tools of the various branches of geometry. An alternative approach is to represent a Euclidean set using functions.

It is therefore appropriate to extend the study of Euclidean sets from geometry to functional analysis by considering their associated functions, such as indicator functions and signed distance functions. This change in the framework of representation allows us to use the concepts and tools of functional analysis in place of those from the various geometric frameworks presented above.

A set problem can thus be mathematically expressed as a functional problem.