This book, "Convex Functions, Monotone Operators and Differentiability" by Robert R. Phelps, is a comprehensive study of the differentiability properties of convex functions on infinite-dimensional spaces. The author provides an in-depth exploration of topics such as convex functions, monotone operators, and differentiable functions, with a focus on Banach spaces. The book covers a wide range of results and contributions from various mathematicians, including D. Preiss, R. Haydon, S. Simons, G. Godefroy, R. Deville, V. Zizler, J. Borwein, S. Fitzpatrick, P. Kenderov, I. Namioka, N. Ribarska, A. and M. E. Verona, and the author himself. Key topics include: 1. **Convex Functions**: Definitions, differentiability properties, and Mazur's theorem on Gâteaux differentiability. 2. **Monotone Operators**: Detailed study of monotone operators, their subdifferentials, and connections to convex functions. 3. **Lower Semicontinuous Convex Functions**: Results on subdifferentials, Ekeland's variational principle, and the Bishop-Phelps theorems. 4. **Smooth Variational Principles**: β-differentiability, smooth bump functions, and the Godefroy-Deville-Zizler variational principles. 5. **Asplund Spaces and the RNP**: Duality between Asplund spaces and spaces with the Radon-Nikodym property (RNP), and perturbed optimization. 6. **Gâteaux Differentiability Spaces**: Equivalence with M-differentiability spaces and stability results. 7. **Usco Maps**: Generalization of monotone operators, topological proofs, and the Clarke subdifferential. The book is structured into seven sections, each delving into specific aspects of the subject, and includes references and an index. It is suitable for graduate students and researchers in functional analysis and related fields.This book, "Convex Functions, Monotone Operators and Differentiability" by Robert R. Phelps, is a comprehensive study of the differentiability properties of convex functions on infinite-dimensional spaces. The author provides an in-depth exploration of topics such as convex functions, monotone operators, and differentiable functions, with a focus on Banach spaces. The book covers a wide range of results and contributions from various mathematicians, including D. Preiss, R. Haydon, S. Simons, G. Godefroy, R. Deville, V. Zizler, J. Borwein, S. Fitzpatrick, P. Kenderov, I. Namioka, N. Ribarska, A. and M. E. Verona, and the author himself. Key topics include: 1. **Convex Functions**: Definitions, differentiability properties, and Mazur's theorem on Gâteaux differentiability. 2. **Monotone Operators**: Detailed study of monotone operators, their subdifferentials, and connections to convex functions. 3. **Lower Semicontinuous Convex Functions**: Results on subdifferentials, Ekeland's variational principle, and the Bishop-Phelps theorems. 4. **Smooth Variational Principles**: β-differentiability, smooth bump functions, and the Godefroy-Deville-Zizler variational principles. 5. **Asplund Spaces and the RNP**: Duality between Asplund spaces and spaces with the Radon-Nikodym property (RNP), and perturbed optimization. 6. **Gâteaux Differentiability Spaces**: Equivalence with M-differentiability spaces and stability results. 7. **Usco Maps**: Generalization of monotone operators, topological proofs, and the Clarke subdifferential. The book is structured into seven sections, each delving into specific aspects of the subject, and includes references and an index. It is suitable for graduate students and researchers in functional analysis and related fields.