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AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS VOLUME XXXI
FUNCTIONAL ANALYSIS AND SEMI-GROUPS BY EINAR HILLE PROFESSOR OF
MATHEMATICS YALE UNIVERSITY PUBLISHED BY THE AMERICAN MATHEMATICAL
SOCIETY 531 WEST 116iH STREET, NEW YORK CITY 1948 To KIRSTI And
each man hears as the twilight nears, to the beat of his dying
hearty The Devil drum on the darkened pane You did it, but was it
Art FOREWORD The analytical theory of semi-groups is a recent
addition to the ever-growing list of mathematical disciplines. It
was my good fortune to take an early interest in this disci pline
and to see it reach maturity. It has been a pleasant association I
hail a semi-group when I see one and I seem to see them every where
Friends have observed, however, that there are mathematical objects
which are not semi-groups. The present book is an elaboration of my
Colloquium Lectures delivered before the American Mathematical
Society at its August, 1944 meeting at Wellesley College. I wish to
thank the Society and its officers for their invitation to present
and publish these lectures. The book is divided into three parts
plus an appendix. My desire to give a practically self-contained
presentation of the theory required the inclusion of an elaborate
introduc tion to modern functional analysis with special emphasis
on function theory in Banach spaces and algebras. This occupies
Part One of the book and the Appendix these portions can be read
separately from the rest and may be used as a text in a course on
operator theory. It is possible to cover most of the material in
these six chapters in two terms. The analytical theory of
one-parameter semi-groups occupies Part Two while Part Three deals
with theapplications to analysis. The latter include such varied
topics as trigonometric series and integrals, summability,
fractional integration, stochastic theory, and the problem of
Cauchy for partial differential equations. In the general theory
the reader will also find an alternate approach to ergodic theory.
All semi-groups studied in this treatise are referred to a normed
topology semi-groups without topology figure in a few places but no
details are given. The task of developing an adequate theory of
trans formation semi-groups operating in partially ordered spaces
is left to more competent hands. The literature has been covered
rather incompletely owing to recent war conditions and to the wide
range of topics touched upon, which have made it exceedingly
difficult to give the proper credits. This investigation has been
supported by grants from the American Philosophical Society and
from Yale University which are gratefully acknowledged. On the
personal side, it is a great pleasure to express my gratitude to
the many friends who have aided me in pre paring this book. J. D.
Tamarkin, who read and criticized my early work in the field and
who vigorously urged its inclusion in the Colloquium Series is
beyond the reach of my grati tude. I am deeply indebted to Nelson
Dunford and to Max Zorn who have contributed extensively to the
book, the former chiefly to Chapters II, III, V, VIII, IX, and XIV,
the latter to Chapters IV, VII, and XXII. Both have given me
generously of their time and special experience. Various portions
of the manuscript have been critically examined and amended by
Warren Ambrose, E. G. Begle, H. Cramdr, J. L. Doob, W. Feller, N.
Jacobson, D. S. Miller, II. Pollard, C.E. Rickart, and I. E. Segal.
To all helpers, named and un named, I extend my warmest thanks.
EINAK HILLE New Haven, Conn., December, 1946 CONVENTIONS Each Part
of the book starts with a Summary, each Chapter with an
Orientation. The chapters are divided into sections and the
sections, except orientations, are grouped into paragraphs. Cross
references are normally to sections, rarely to paragraphs. Section
3.10 is the tenth section of Chapter III it belongs to 2 which is
referred to as 3.2 when necessary...
AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS VOLUME XXXI
FUNCTIONAL ANALYSIS AND SEMI-GROUPS BY EINAR HILLE PROFESSOR OF
MATHEMATICS YALE UNIVERSITY PUBLISHED BY THE AMERICAN MATHEMATICAL
SOCIETY 531 WEST 116iH STREET, NEW YORK CITY 1948 To KIRSTI And
each man hears as the twilight nears, to the beat of his dying
hearty The Devil drum on the darkened pane You did it, but was it
Art FOREWORD The analytical theory of semi-groups is a recent
addition to the ever-growing list of mathematical disciplines. It
was my good fortune to take an early interest in this disci pline
and to see it reach maturity. It has been a pleasant association I
hail a semi-group when I see one and I seem to see them every where
Friends have observed, however, that there are mathematical objects
which are not semi-groups. The present book is an elaboration of my
Colloquium Lectures delivered before the American Mathematical
Society at its August, 1944 meeting at Wellesley College. I wish to
thank the Society and its officers for their invitation to present
and publish these lectures. The book is divided into three parts
plus an appendix. My desire to give a practically self-contained
presentation of the theory required the inclusion of an elaborate
introduc tion to modern functional analysis with special emphasis
on function theory in Banach spaces and algebras. This occupies
Part One of the book and the Appendix these portions can be read
separately from the rest and may be used as a text in a course on
operator theory. It is possible to cover most of the material in
these six chapters in two terms. The analytical theory of
one-parameter semi-groups occupies Part Two while Part Three deals
with theapplications to analysis. The latter include such varied
topics as trigonometric series and integrals, summability,
fractional integration, stochastic theory, and the problem of
Cauchy for partial differential equations. In the general theory
the reader will also find an alternate approach to ergodic theory.
All semi-groups studied in this treatise are referred to a normed
topology semi-groups without topology figure in a few places but no
details are given. The task of developing an adequate theory of
trans formation semi-groups operating in partially ordered spaces
is left to more competent hands. The literature has been covered
rather incompletely owing to recent war conditions and to the wide
range of topics touched upon, which have made it exceedingly
difficult to give the proper credits. This investigation has been
supported by grants from the American Philosophical Society and
from Yale University which are gratefully acknowledged. On the
personal side, it is a great pleasure to express my gratitude to
the many friends who have aided me in pre paring this book. J. D.
Tamarkin, who read and criticized my early work in the field and
who vigorously urged its inclusion in the Colloquium Series is
beyond the reach of my grati tude. I am deeply indebted to Nelson
Dunford and to Max Zorn who have contributed extensively to the
book, the former chiefly to Chapters II, III, V, VIII, IX, and XIV,
the latter to Chapters IV, VII, and XXII. Both have given me
generously of their time and special experience. Various portions
of the manuscript have been critically examined and amended by
Warren Ambrose, E. G. Begle, H. Cramdr, J. L. Doob, W. Feller, N.
Jacobson, D. S. Miller, II. Pollard, C.E. Rickart, and I. E. Segal.
To all helpers, named and un named, I extend my warmest thanks.
EINAK HILLE New Haven, Conn., December, 1946 CONVENTIONS Each Part
of the book starts with a Summary, each Chapter with an
Orientation. The chapters are divided into sections and the
sections, except orientations, are grouped into paragraphs. Cross
references are normally to sections, rarely to paragraphs. Section
3.10 is the tenth section of Chapter III it belongs to 2 which is
referred to as 3.2 when necessary...
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