Elements of Cartesian Analysis in Saidou Spaces

Abstract EN

mong the various aspects of mathematics that have contributed to functional analysis, we must underline the convex analysis, a branch that has had fertile tracks in analytical and even geometric forms.It will be shown that convex analysis and Cartesian analysis coincide in particular cases, but generally they are different.

We have to denote that this geometry will be a generalization of the polyhedron and polytopes in affine spaces, see[5],[6].

The regular polytopes and polyhedron are convex sets but cartesian sets are the union not necessery convex of manuy polytops see Figure4 below.

After defining and characterizing this new notion, you will realize how this track would open interesting doors in the field of the applications of mathematics for the benefit of applied sciences, like medicine, physics, engineering,economics and even sociology.We will give an original definition of a Cartesian set and then to define a locally cartesian topology whichcalled Saidou Space.

At the last part of this paper we start the main studie of this paper which is to define a new kind of functions and caracterize the relationship between the cartesian topology and the cartesianfunctions by doing a functional analysis.

This is to prepare the elements for the optimization of cartesianproblems for futur works.