The article discusses the concepts of proximity and duality in Hilbert spaces, focusing on the projection of a point onto a convex closed subset and the extension of this idea to functions. It introduces the notion of dual functions and their properties, leading to the concept of proximal mappings. The key result is that any element in a Hilbert space can be uniquely decomposed into two elements, one from each of two dual convex functions, which are orthogonal. The paper also explores the properties of these dual functions, their relationship with convex functions, and the implications for optimization and functional analysis. It includes theorems on the existence and uniqueness of proximal mappings, the contraction of distances, and the characterization of proximal mappings through their dual functions. The study also addresses the relationship between dual functions and their conjugates, as well as the properties of functions that are more or less convex than the quadratic function. The paper concludes with a discussion on the characterization of proximal mappings and their applications in functional analysis.The article discusses the concepts of proximity and duality in Hilbert spaces, focusing on the projection of a point onto a convex closed subset and the extension of this idea to functions. It introduces the notion of dual functions and their properties, leading to the concept of proximal mappings. The key result is that any element in a Hilbert space can be uniquely decomposed into two elements, one from each of two dual convex functions, which are orthogonal. The paper also explores the properties of these dual functions, their relationship with convex functions, and the implications for optimization and functional analysis. It includes theorems on the existence and uniqueness of proximal mappings, the contraction of distances, and the characterization of proximal mappings through their dual functions. The study also addresses the relationship between dual functions and their conjugates, as well as the properties of functions that are more or less convex than the quadratic function. The paper concludes with a discussion on the characterization of proximal mappings and their applications in functional analysis.