Elementary Functional Analysis (De Gruyter Textbook) by Marat V. Markin (PDF)

Description

While there is a plethora of excellent, but mostly “tell-it-all” books on the subject, this one is intended to take a unique place in what today seems to be a still wide open niche for an introductory text on the basics of functional analysis to be taught within the existing constraints of the standard, for the United States, one-semester graduate curriculum (fifteen weeks with two seventy-five-minute lectures per week). The book consists of seven chapters and an appendix taking the reader from the fundamentals of abstract spaces (metric, vector, normed vector, and inner product), through the basics of linear operators and functionals, the three fundamental principles (the Hahn-Banach Theorem, the Uniform Boundedness Principle, the Open Mapping Theorem and its equivalents: the Inverse Mapping and Closed Graph Theorems) with their numerous profound implications and certain interesting applications, to the elements of the duality and reflexivity theory. Chapter 1 outlines some necessary preliminaries, while the Appendix gives a concise discourse on the celebrated Axiom of Choice, its equivalents (the Hausdorff Maximal Principle, Zorn’s Lemma, and Zermello’s Well-Ordering Principle), and ordered sets. Being designed as a text to be used in a classroom, the book constantly calls for the student’s actively mastering the knowledge of the subject matter. It contains 112 Problems, which are indispensable for understanding and moving forward. Many important statements are given as problems, a lot of these are frequently referred to and used in the main body. There are also 376 Exercises throughout the text, including Chapter 1 and the Appendix, which require of the student to prove or verify a statement or an example, fill in necessary details in a proof, or provide an intermediate step or a counterexample. They are also an inherent part of the material. More difficult problems are marked with an asterisk, many problem and exercises being supplied with “existential” hints. The book is generous on Examples and contains numerous Remarks accompanying every definition and virtually each statement to discuss certain subtleties, raise questions on whether the converse assertions are true, whenever appropriate, or whether the conditions are essential. The prerequisites are set intentionally quite low, the students not being assumed to have taken graduate courses in real or complex analysis and general topology, to make the course accessible and attractive to a wider audience of STEM (science, technology, engineering, and mathematics) graduate students or advanced undergraduates with a solid background in calculus and linear algebra. Withproper attention given to applications, plenty of examples, problems, and exercises, this well-designed text is ideal for a one-semester graduate course on the fundamentals of functional analysis for students in mathematics, physics, computer science, and engineering. ContentsPreliminariesMetric SpacesNormed Vector and Banach SpacesInner Product and Hilbert SpacesLinear Operators and FunctionalsThree Fundamental Principles of Linear Functional AnalysisDuality and ReflexivityThe Axiom of Choice and Equivalents

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Elementary Functional Analysis is a very thorough, systematic, and logically structured introduction the the rather challenging field of functional analysis. The book assumes little on the part of the reader beyond a knowledge of Linear Algebra and a strong interest in the topic.Exercises for the reader are mixed in along the way; for a Individual reader I highly recommend solving them as you go, in a course, the instructor would do well to cover the more difficult of them in-class. The problems found at the end of the chapter are quite challenging, but proceed in a logical manner, building in difficulty to help the reader develop the skills to meet their challenge.The first chapter lists certain preliminary information that should be known to the reader before beginning. The second chapter is a thorough introduction to metric space theory, more complete than found in most topology texts, and with a particular emphasis on completeness, essential to a real understanding of the topic. Chapter three consists of a short review of vector spaces, then delves into Normed and Banach spaces. Chapter four introduces the inner product, explains it’s geometry, and then describes Hilbert Spaces.Chapters 5, 6, and 7 are the real meat of the book. Chapter 5 introduces the reader to the relevant points of operator theory, and in particular, the linear functionals, setting up the material needed for chapter 6, in which we are introduced to the three fundamental principals: The open mapping theorem, Hahn-Banach, and the Uniform Boundedness principle. The problems at the end of chapter 6 illustrate why these three principles are considered the cornerstone of linear functional analysis.Chapterr 7 introduces duality, convergence in the weak and weak* topologies, and and finally, and essentially, reflexivity, giving the reader the essential tools of contemporary functional analysis, making them well prepared to begin a more advanced examination of the topic, or even their own research.All in all, it is a well-constructed and thorough introduction, rendering a very difficult topic much more accessible.

⭐

⭐

Keywords

Free Download Elementary Functional Analysis (De Gruyter Textbook) in PDF format
Elementary Functional Analysis (De Gruyter Textbook) PDF Free Download
Download Elementary Functional Analysis (De Gruyter Textbook) 2018 PDF Free
Elementary Functional Analysis (De Gruyter Textbook) 2018 PDF Free Download
Download Elementary Functional Analysis (De Gruyter Textbook) PDF
Free Download Ebook Elementary Functional Analysis (De Gruyter Textbook)