This paper, authored by Stefan Banach, explores the operations in abstract sets and their application to integral equations. The introduction defines an operation as a unique relation \( yRx \) such that if \( yRx \) and \( zRx \), then \( y = z \). The paper discusses the concept of functional or line functions, which are operations with domains and ranges that are sets of functions. It highlights the contributions of mathematicians like Volterra, Fréchet, Hadamard, Riesz, Pincherle, Steinhaus, and Weil to the study of these functions. Banach introduces the notion of a sphere in an abstract set \( E \) and defines key concepts such as convergence, continuity, and limits in this context. He proves several theorems, including: 1. **Theorem 1**: The triangle inequality for norms. 2. **Theorem 2**: The reverse triangle inequality for norms. 3. **Theorem 3**: A bound on the norm of a sum of elements. 4. **Theorem 4**: Uniqueness of limits. 5. **Theorem 5**: Convergence of norms implies convergence of the sequence. 6. **Theorem 6**: Continuity of operations. 7. **Theorem 7**: Convergence of a sequence of spheres. 8. **Theorem 8**: Convergence of series. 9. **Theorem 9**: Conditions for convergence of a sequence. 10. **Theorem 10**: Conditions for a sphere to contain another sphere. 11. **Theorem 11**: Convergence of a sequence of spheres. 12. **Theorem 12**: Closure of a set. 13. **Theorem 13**: Existence of a sphere disjoint from a set. The paper also discusses additive operations and their properties, including: 1. **Theorem 1**: Continuity of bounded additive operations. 2. **Theorem 2**: Continuity of additive operations. 3. **Theorem 3**: Continuity of additive operations under certain conditions. 4. **Theorem 4**: Continuity of additive operations under convergence conditions. 5. **Theorem 6**: Existence of a fixed point for continuous operations. 6. **Theorem 7**: Solution of equations involving additive operations. Banach concludes by extending these concepts to measurable functions, defining operations and norms in this context. The paper aims to establish general theorems applicable to various functional fields, providing a foundation for further research in abstract set theory and functional analysis.This paper, authored by Stefan Banach, explores the operations in abstract sets and their application to integral equations. The introduction defines an operation as a unique relation \( yRx \) such that if \( yRx \) and \( zRx \), then \( y = z \). The paper discusses the concept of functional or line functions, which are operations with domains and ranges that are sets of functions. It highlights the contributions of mathematicians like Volterra, Fréchet, Hadamard, Riesz, Pincherle, Steinhaus, and Weil to the study of these functions. Banach introduces the notion of a sphere in an abstract set \( E \) and defines key concepts such as convergence, continuity, and limits in this context. He proves several theorems, including: 1. **Theorem 1**: The triangle inequality for norms. 2. **Theorem 2**: The reverse triangle inequality for norms. 3. **Theorem 3**: A bound on the norm of a sum of elements. 4. **Theorem 4**: Uniqueness of limits. 5. **Theorem 5**: Convergence of norms implies convergence of the sequence. 6. **Theorem 6**: Continuity of operations. 7. **Theorem 7**: Convergence of a sequence of spheres. 8. **Theorem 8**: Convergence of series. 9. **Theorem 9**: Conditions for convergence of a sequence. 10. **Theorem 10**: Conditions for a sphere to contain another sphere. 11. **Theorem 11**: Convergence of a sequence of spheres. 12. **Theorem 12**: Closure of a set. 13. **Theorem 13**: Existence of a sphere disjoint from a set. The paper also discusses additive operations and their properties, including: 1. **Theorem 1**: Continuity of bounded additive operations. 2. **Theorem 2**: Continuity of additive operations. 3. **Theorem 3**: Continuity of additive operations under certain conditions. 4. **Theorem 4**: Continuity of additive operations under convergence conditions. 5. **Theorem 6**: Existence of a fixed point for continuous operations. 6. **Theorem 7**: Solution of equations involving additive operations. Banach concludes by extending these concepts to measurable functions, defining operations and norms in this context. The paper aims to establish general theorems applicable to various functional fields, providing a foundation for further research in abstract set theory and functional analysis.