首先，本书讲清楚了泛函分析理论对数学其他领域的应用。其次，本书讲清楚了分析理论在诸多领域（如物理学、化学、生物学、工程技术和经济学等等）的广泛应用。再次，本书由浅人深地讲透了基本理论的发展历程及走向，它既讲清楚了所涉及学科的具体问题，也讲清楚了其背后的数学原理及其作用。

\"Preface Translator\'s Preface INTRODUCTION Mathematics and Physics APPLICATIONS IN MECHANICS CHAPTER 58 Basic Equations of Point Mechanics 58.1. Notations 58.2. Lever Principle and Stability of the Scales 58.3. Perspectives 58.4. Kepler\'s Laws and a Look at the History of Astronomy 58.5. Newton\'s Basic Equations 58.6. Changes of the System of Reference and the Role of Inertial Systems 58.7. General Point System and Its Conserved Quantities 58.8. Newton\'s Law of Gravitation and Coulomb\'s Law of Electrostatics 58.9. Application to the Motion of Planets 58.10. Gauss\' Principle of Least Constraint and the General Basic Equations of Point Mechanics with Side Conditions 58.11. Principle of Virtual Power 58.12. Equilibrium States and a General Stability Principle 58.13. Basic Equations of the Rigid Body and the Main Theorem about the Motion of the Rigid Body and Its Equilibrium 58.14. Foundation of the Basic Equations of the Rigid Body 58.15. Physical Models, the Expansion of the Universe, and Its Evolution after the Big Bang 58.16. Legendre Transformation and Conjugate Functionals 58.17. Lagrange Multipliers 58.18. Principle of Stationary Action 58.19. Trick of Position Coordinates and Lagrangian Mechanics 58.20. Hamiltonian Mechanics 58.21. Poissonian Mechanics and Heisenberg\'s Matrix Mechanics in Quantum Theory 58.22. Propagation of Action 58.23. Hamilton-Jacobi Equation 58.24. Canonical Transformations and the Solution of the Canonical Equations via the Hamilton-Jacobi Equation 58.25. Lagrange Brackets and the Solution of the Hamilton-Jacobi Equation via the Canonical Equations 58.26. Initial-Value Problem for the Hamiiton-Jacobi Equation 58.27. Dimension Analysis CHAPTER 59 Dualism Between Wave and Particle, Preview of Quantum Theory, and Elementary Particles 59.1. Plane Waves 59.2. Polarization 59.3. Dispersion Relations 59.4. Spherical\"