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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.
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