Functional Analysis

1. Front Matter

   Pages I-XII

   PDF

2. Preliminaries

   • Kôsaku Yosida

   Pages 1-22

3. Semi-norms

   • Kôsaku Yosida

   Pages 23-68

4. Applications of the Baire-Hausdorff Theorem

   • Kôsaku Yosida

   Pages 68-81

5. The Orthogonal Projection and F. Riesz’ Representation Theorem

   • Kôsaku Yosida

   Pages 81-101

6. The Hahn-Banach Theorems

   • Kôsaku Yosida

   Pages 102-119

7. Strong Convergence and Weak Convergence

   • Kôsaku Yosida

   Pages 119-145

8. Fourier Transform and Differential Equations

   • Kôsaku Yosida

   Pages 145-193

9. Dual Operators

   • Kôsaku Yosida

   Pages 193-208

10. Resolvent and Spectrum

    • Kôsaku Yosida

    Pages 209-231

11. Analytical Theory of Semi-groups

    • Kôsaku Yosida

    Pages 231-274

12. Compact Operators

    • Kôsaku Yosida

    Pages 274-293

13. Normed Rings and Spectral Representation

    • Kôsaku Yosida

    Pages 294-362

14. Other Representation Theorems in Linear Spaces

    • Kôsaku Yosida

    Pages 362-378

15. Ergodic Theory and Diffusion Theory

    • Kôsaku Yosida

    Pages 379-418

16. The Integration of the Equation of Evolution

    • Kôsaku Yosida

    Pages 418-445

17. Back Matter

    Pages 447-476

    PDF

The present book is based on lectures given by the author at the University of Tokyo during the past ten years. It is intended as a textbook to be studied by students on their own or to be used in a course on Functional Analysis, i. e. , the general theory of linear operators in function spaces together with salient features of its application to diverse fields of modern and classical analysis. Necessary prerequisites for the reading of this book are summarized, with or without proof, in Chapter 0 under titles: Set Theory, Topo­ logical Spaces, Measure Spaces and Linear Spaces. Then, starting with the chapter on Semi-norms, a general theory of Banach and Hilbert spaces is presented in connection with the theory of generalized functions of S. L. SOBOLEV and L. SCHWARTZ. While the book is primarily addressed to graduate students, it is hoped it might prove useful to research mathe­ maticians, both pure and applied. The reader may pass, e. g. , from Chapter IX (Analytical Theory of Semi-groups) directly to Chapter XIII (Ergodic Theory and Diffusion Theory) and to Chapter XIV (Integration of the Equation of Evolution). Such materials as "Weak Topologies and Duality in Locally Convex Spaces" and "Nuclear Spaces" are presented in the form of the appendices to Chapter V and Chapter X, respectively. These might be skipped for the first reading by those who are interested rather in the application of linear operators.

• Analysis
• Hilbert space
• calculus
• differential equation
• functional analysis

• Research Institute for Mathematical Sciences, University of Kyoto, Kyoto, Japan

  Kôsaku Yosida

Policies and ethics