Kosaku Yosida, born on February 7, 1909, was brought up in Tokyo. Having majored in Mathematics at University of Tokyo, he was appointed to Assistant at Osaka University in 1933 and promoted to Associate Professor in 1934. He re ceived the title of Doctor of Science from Osaka University in 1939. In 1942 he was appointed to Professor at Nagoya University, where he worked very hard with his colleagues to promote and expand the newly established Department of Mathe matics. He was appointed to Professor at Osaka University in 1953 and then to Professor at University of Tokyo in 1955. After retiring from University of Tokyo in 1969, he was appointed to Professor at Kyoto University, where he also acted as Director of the Research Institute for Mathematical Sciences. He retired from Kyoto University in 1972 and worked as Professor at Gakushuin University until 1979. Yosida acted as President of the Mathematical Society of Japan, as Member of the Science Council of Japan, and as Member of the Executive Committee of the International Mathematical Union. In 1967 he received the Japan Academy Prize and the Imperial Prize for his famous work on the theory of semigroups and its applications. In 1971 he was elected Member of the Japan Academy. Yosida went abroad many times to give series of lectures at mathematical in stitutions and to deliver invited lectures at international mathematical symposia.

Teiji Takagi one of the leading number theorists of this century, is most renowned as the founder of class field theory. This volume reflects the stages of his development of this theory. Inspired by a genial idea related to analytic number theory, he developed a beautiful general theory of abelian extensions of algebraic number fields which he addressed at the ICM 1920 at Strasbourg. This report ends with a problem to generalize the results to the case of normal, not necessarily abelian extensions. Up to now this problem has stimulated research. This second edition incorporates the whole contents of "The Collected Papers of Teiji Takagi" edited by S. Kuroda, published by Iwanami Shoten in 1974. Following additions have been made: Note on Eulerian squares (1946).- Concept of numbers.- K. Iwasawa: On arithmetical papers of Takagi.- K. Yosida: On analytical papers of Takagi.- S. Iyanaga: On life and works of Takagi.

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.

In the end of the last century, Oliver Heaviside inaugurated an operational calculus in connection with his researches in electromagnetic theory. In his operational calculus, the operator of differentiation was denoted by the symbol "p". The explanation of this operator p as given by him was difficult to understand and to use, and the range of the valid ity of his calculus remains unclear still now, although it was widely noticed that his calculus gives correct results in general. In the 1930s, Gustav Doetsch and many other mathematicians began to strive for the mathematical foundation of Heaviside's operational calculus by virtue of the Laplace transform -pt e f(t)dt. ( However, the use of such integrals naturally confronts restrictions con cerning the growth behavior of the numerical function f(t) as t ~ ~. At about the midcentury, Jan Mikusinski invented the theory of con volution quotients, based upon the Titchmarsh convolution theorem: If f(t) and get) are continuous functions defined on [O,~) such that the convolution f~ f(t-u)g(u)du =0, then either f(t) =0 or get) =0 must hold. The convolution quotients include the operator of differentiation "s" and related operators. Mikusinski's operational calculus gives a satisfactory basis of Heaviside's operational calculus; it can be applied successfully to linear ordinary differential equations with constant coefficients as well as to the telegraph equation which includes both the wave and heat equa tions with constant coefficients.