本书是一部数学经典教材，初版于1965年，以作者在东京大学任教十余年所用的讲义为基础写成的。经过几次修订和增补，1980年出了第5版，本版（第6版）实际上是第5版的重印版。全书论述了泛函空间的线性算子理论及其在现代分析和经典分析各领域中的许多应用。目次：预备知识；半范数；Baire-Hausdorff定理的应用；正交射影和riesz表示定理；Hahn-Banach定理；强收敛和弱收敛；傅里叶变换和微分方程；对偶算子；预解和谱；半群的解析理论；紧致算子；赋范环和谱表示；线性空间中的其他表示定理；遍历性理论和扩散理论；发展方程的积分。读者对象：数学专业的研究生和科研人员。

   0.Preliminaries

1.Set Theory

2.Topological Spaces

3.Measure Spaces

4.Linear Spaces

I.Semi-norms

1.Semi-norms and Locally Convex Linear Topological Spaces

2.Norms and Quasi-norms

3.Examples of Normed Linear Spaces

4.Examples of Quasi-normed Linear Spaces

5.Pre-Hilbert Spaces

6.Continuity of Linear Operators

7.Bounded Sets and Bornologic Spaces

8.Generalized Functions and Generalized Derivatives

9.B-spaces and F-spaces

10.The Completion

11.Factor Spaces of a B-space

12.The Partition of Unity

13.Generalized Functions with Compact Support

14.The Direct Product of Generalized Functions

II.Applications of the Baire-Hausdorff Theorem

1.The Uniform Boundedness Theorem and the Resonance Theorem

2.The Vitali-Hahn-Saks Theorem

3.The Termwise Differentiability of a Sequence of Generalized Functions

4.The Principle of the Condensation of Singularities

5.The Open Mapping Theorem

6.The Closed Graph Theorem

7.An Application of the Closed Graph Theorem (Hormander's Theorem)

III.The Orthogonal Projection and F.Riesz' Representation Theorem

1.The Orthogonal Projection

2."Nearly Orthogonal" Elements

3.The Ascoli-Arzela Theorem

4.The Orthogonal Base.Bessel's Inequality and Parseval's Relation

5.E.Schmidt's Orthogonalization

6.F.Riesz' Representation Theorem

7.The Lax-Milgram Theorem

8.A Proof of the Lebesgue-Nikodym Theorem

9.The Aronszajn-Bergman Reproducing Kernel

10.The Negative Norm of P.LAX

11.Local Structures of Generalized Functions

IV.The Hahn-Banach Theorems

1.The Hahn-Banach Extension Theorem in Real Linear Spaces

2.The Generalized Limit

3.Locally Convex, Complete Linear Topological Spaces

4.The Hahn-Banach Extension Theorem in Complex Linear Spaces

5.The Hahn-Banach Extension Theorem in Normed Linear Spaces

6.The Existence of Non-trivial Continuous Linear Functionals

7.Topologies of Linear Maps

8.The Embedding of X in its Bidual Space X"

9.Examples of Dual Spaces

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吉田耕作是东京大学教授。