This text is a detailed mathematical paper by Stefan Banach, published in *Fundamenta Mathematicae* III, focusing on operations in abstract sets and their applications to integral equations. The paper introduces the concept of functional operations, defines axioms for algebraic structures, and establishes theorems related to norms, limits, and continuity in function spaces. It discusses the properties of continuous operations, the convergence of sequences and series in function spaces, and the concept of boundedness and uniform continuity. The paper also explores the theory of additive operations and their continuity, as well as the existence of fixed points for certain types of operations. The work is foundational in functional analysis and provides a rigorous framework for understanding the behavior of functions in abstract spaces.This text is a detailed mathematical paper by Stefan Banach, published in *Fundamenta Mathematicae* III, focusing on operations in abstract sets and their applications to integral equations. The paper introduces the concept of functional operations, defines axioms for algebraic structures, and establishes theorems related to norms, limits, and continuity in function spaces. It discusses the properties of continuous operations, the convergence of sequences and series in function spaces, and the concept of boundedness and uniform continuity. The paper also explores the theory of additive operations and their continuity, as well as the existence of fixed points for certain types of operations. The work is foundational in functional analysis and provides a rigorous framework for understanding the behavior of functions in abstract spaces.