tailieunhanh - Ebook Principles of Real Analysis (Third Edition): Part 2

(BQ) The book is largely about the Lebesgue theory of integration, but includes a very thorough coverage of the theory of metric and topological spaces in the first two chapters. Chapters 3,4 and 5 are the heart of the book covering measure theory, the Lebesgue integral and some topics from introductory functional analysis like theory of operators and Banach spaces. Chapters 6 and 7, covering Hibert spaces, the Radon Nikodym theorem and the Riesz Representation Theorem among other things, are the most useful for someone like me who wants to master higher analysis in order to read financial mathematics. And what's more, there is a solutions book providing answers to all 609 problems (spread over 7 chapters!) and more. All in all, the authors have made a great contribution! | CHAPTER 5 NORMED SPACES AND Lp-SPACES The algebraic theory of vector spaces has been an integral part of modem mathematics for some time. In analysis one studies vector spaces from the topological point of view taking into consideration the already existing algebraic structure. The most fruitful study comes when one attaches to every vector a real number called the norm of the vector. The norm can be thought of as a generalization of the concept of length. A normed space that is complete in the metric induced by its norm is called a Banach1 space. A variety of diverse problems from different branches of mathematics and science in general can be translated to the framework of Banach space theory and solved by applying its powerful techniques. For this reason the theory of Banach spaces is in the frontier of modem mathematical research. This chapter presents a brief introduction to the theory of normed and Banach spaces. After developing the basic properties of normed spaces the three cornerstones of functional analysis are proved the principle of uniform boundedness the open mapping theorem and the Hahn-Banach Then a section is devoted to the study of Banach lattices that is Banach spaces whose norms are compatible with the lattice structure of the spaces. As we shall see many Banach lattices are actually old friends. Finally the classical Zp-spaces are investigated and the theory of integration is placed in its appropriate perspective. 27. NORMED SPACES AND BANACH SPACES A real-valued function II ll defined on a vector space X is called a norm if it satisfies the following three properties Stefan Banach 1892-1945 a prominent Polish mathematician. He is the founder of the contemporary field of functional analysis. 2Hans Hahn 1879-1934 an Austrian mathematician and philosopher. He contributed decisively to functional analysis general topology and the foundations of mathematics. 217 218 Chapter 5 NORMED SPACES AND Lp-SPACES 1- IIa II 0 for each A e X and a .

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