This paper, authored by Jean-Jacques Moreau, explores the concepts of proximity and duality in a Hilbert space. The author begins by introducing the notion of projection onto a convex closed set and extends this idea to functions in the space of convex functions. The key result is that any point in the Hilbert space can be decomposed into the sum of a point in the set and a point in its dual set, with orthogonality conditions. This decomposition is generalized to functions, where the function \( f \) and its dual function \( g \) are related through the inequality \( f(x) + g(y) \geq (x \mid y) \). The paper also discusses the properties of these functions, including their differentiability and the contraction of distances. Additionally, it provides characterizations of functions that are less convex than a given function and the conditions under which a function is the primitive of a proximal mapping. The paper concludes with a discussion on sur-dual functions and their applications in functional analysis.This paper, authored by Jean-Jacques Moreau, explores the concepts of proximity and duality in a Hilbert space. The author begins by introducing the notion of projection onto a convex closed set and extends this idea to functions in the space of convex functions. The key result is that any point in the Hilbert space can be decomposed into the sum of a point in the set and a point in its dual set, with orthogonality conditions. This decomposition is generalized to functions, where the function \( f \) and its dual function \( g \) are related through the inequality \( f(x) + g(y) \geq (x \mid y) \). The paper also discusses the properties of these functions, including their differentiability and the contraction of distances. Additionally, it provides characterizations of functions that are less convex than a given function and the conditions under which a function is the primitive of a proximal mapping. The paper concludes with a discussion on sur-dual functions and their applications in functional analysis.