经典巴拿赫空间Ⅰ和Ⅱ
2010-1
世界图书出版公司
林登斯托斯
242
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The appearance of Banachs book [8] in 1932 signified the beginning of a systematic study of normed linear spaces， which have been the subject of continuous research ever since. In the sixties， and especially in the last decade， the research activity in this area grew considerably. As a result， Banach space theory gained very much in depth as well as in scope. Most of its well known classical problems were solved， many interesting new directions were developed， and deep connections between Banach space theory and other areas of mathematics were established. The purpose of this book is to present the main results and current research directions in the geometry of Banach spaces， with an emphasis on the study of the structure of the classical Banach spaces， that is C（K） and Lp（） and related spaces.We did not attempt to write a comprehensive survey of Banach space theory， or even only of the theory of classical Banach spaces， since the amount of interesting results on the subject makes such a survey practically impossible. A part of the subject matter of this book appeared in outline in our lecture notes[96]. In contrast to those notes， most of the results presented here are given with complete proofs. We therefore hope that it will be possible to use the present book both as a text book on Banach space theory and as a reference book for research workers in the area. It contains much material which was not discussed in [96]， a large part of which being the result of very recent research work. An indication to the rapid recent progress in Banach space theory is the fact that most of the many problems stated in [96] have been solved by now. In the present volume we also state some open problems. It is reasonable to expect that many of these will be solved in the not too far future. We feel， however，that most of the topics discussed here have reached a relatively final form， and that their presentation will not be radically affected by the solution of the open problems. Among the topics discussed in detail in this volume， the one which seems to us to be the least well understood and which might change the most in the future， is that of the approximation property.
本书是Springer数学经典教材之一。本书延续了该系列书的一贯风格，深入但不深沉。材料新颖，许多内容是同类书籍不具备的。对于学习Banach空间结构理论的学者来说，这是一本参考价值极高的书籍；对于学习该科目的读者，本书也是同等重要。目次：schauder 基；C0空间和lp空间；对称基；O rlicz序列空间。 读者对象：数学专业高年级的学生、老师和相关的科研人员。
1. Schauder Bases a. Existence of Bases and Examples b. Schauder Bases and Duality c. Unconditional Bases d. Examples of Spaces Without an Unconditional Basis e. The Approximation Property f. Biorthogonal Systems g. Schauder Decompositions 2. The Spaces co and lp a. Projections in co and lp and Characterizations of these Spaces b. Absolutely Summing Operators and Uniqueness of Unconditional Bases c. Fredholm Operators, Strictly Singular Operators and Complemented Subspaces of lp lr d. Subspaces of Co and lp and the Approximation Property, Complementably Universal Spaces e. Banach Spaces Containing Iv or co f. Extension and Lifting Properties, Automorphisms of loo, co and lx 3. Symmetric Bases a. Properties of Symmetric Bases, Examples and Special Block Bases b. Subspaces of Spaces with a Symmetric Basis 4. Orlicz Sequence Spaces a. Subspaces of Orlicz Sequence Spaces which have a Symmetric Basis b. Duality and Complemented Subspaces c. Examples of Orlicz Sequence Spaces. d. Modular Sequence Spaces and Subspaces of Ip lr e. Lorentz Sequence Spaces References Subject Index
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