Douglas S. Bridges (auth.)0387982396
The core of this book, Chapters 3 through 5, presents a course on metric, normed,andHilbertspacesatthesenior/graduatelevel. Themotivationfor each of these chapters is the generalisation of a particular attribute of the n Euclidean spaceR : in Chapter 3, that attribute isdistance; in Chapter 4, length; and in Chapter 5, inner product. In addition to the standard topics that, arguably, should form part of the armoury of any graduate student in mathematics, physics, mathematical economics, theoretical statistics,. . . , this part of the book contains many results and exercises that are seldom found in texts on analysis at this level. Examples of the latter are Wong’s Theorem(3. 3. 12)showingthattheLebesguecoveringpropertyisequivalent to the uniform continuity property, and Motzkin’s result (5. 2. 2) that a nonempty closed subset of Euclidean space has the unique closest point property if and only if it is convex. The sad reality today is that, perceiving them as one of the harder parts oftheirmathematicalstudies,studentscontrivetoavoidanalysiscoursesat almost any cost, in particular that of their own educational and technical deprivation. Many universities have at times capitulated to the negative demand of students for analysis courses and have seriously watered down their expectations of students in that area. As a result, mathematics – jors are graduating, sometimes with high honours, with little exposure to anything but a rudimentary course or two on real and complex analysis, often without even an introduction to the Lebesgue integral. |
Table of contents :
Introduction….Pages 1-8
Front Matter….Pages 9-9
Analysis on the Real Line….Pages 11-77
Differentiation and the Lebesgue Integral….Pages 79-122
Front Matter….Pages 123-123
Analysis in Metric Spaces….Pages 125-171
Analysis in Normed Linear Spaces….Pages 173-231
Hilbert Spaces….Pages 233-258
An Introduction to Functional Analysis….Pages 259-290 |
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