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Über dieses Buch

This text is a self-contained introduction to the three main families that we encounter in analysis – metric spaces, normed spaces, and inner product spaces – and to the operators that transform objects in one into objects in another. With an emphasis on the fundamental properties defining the spaces, this book guides readers to a deeper understanding of analysis and an appreciation of the field as the “science of functions.”

Many important topics that are rarely presented in an accessible way to undergraduate students are included, such as unconditional convergence of series, Schauder bases for Banach spaces, the dual of ℓp topological isomorphisms, the Spectral Theorem, the Baire Category Theorem, and the Uniform Boundedness Principle. The text is constructed in such a way that instructors have the option whether to include more advanced topics.

Written in an appealing and accessible style, Metrics, Norms, Inner Products, and Operator Theory is suitable for independent study or as the basis for an undergraduate-level course. Instructors have several options for building a course around the text depending on the level and interests of their students.

Key features:

Aimed at students who have a basic knowledge of undergraduate real analysis. All of the required background material is reviewed in the first chapter.Suitable for undergraduate-level courses; no familiarity with measure theory is required.Extensive exercises complement the text and provide opportunities for learning by doing.A separate solutions manual is available for instructors via the Birkhäuser website (www.springer.com/978-3-319-65321-1). Unique text providing an undergraduate-level introduction to metrics, norms, inner products, and their associated operator theory.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Notation and Preliminaries

This preliminary chapter is a quick-reference guide to the notation, terminology, and background information that will be assumed throughout the volume. Unlike the rest of the text, in this preliminary chapter we may state results without proof or without the motivation and discussion that is provided throughout the later chapters. Proofs of the facts reviewed in Sections 1.1–1.9 can be found in most calculus texts (such as [HHW18]), or in undergraduate analysis texts (such as [Rud76]).
Christopher Heil

Chapter 2. Metric Spaces

Much of what we do in real analysis centers on issues of convergence or approximation. What does it mean for one object to be close to (or to approximate) another object? How can we define the limit of a sequence of objects that appear to be converging in some sense?
Christopher Heil

Chapter 3. Norms and Banach Spaces

We studied metric spaces in detail in Chapter 2. A metric on a set X provides us with a notion of the distance between points in X. In this chapter we will study norms, which are special types of metrics. However, in order for us to be able to to define a norm, our set X must be a vector space.
Christopher Heil

Chapter 4. Further Results on Banach Spaces

In this chapter we will take a closer look at normed and Banach spaces. In a generic metric space there need not be any way to “add” points in the space, but a normed space is a vector space, so we can form linear combinations of vectors. Moreover, because we also have a notion of limits we can go even further and define infinite series of vectors (which are limits of partial sums of the series).
Christopher Heil

Chapter 5. Inner Products and Hilbert Spaces

In a normed vector space each vector has an assigned length, and from this we obtain the distance from \( x \) to \( y \) as the length of the vector \( x \) \( - \) \( y \).
Christopher Heil

Chapter 6. Operator Theory

Our focus in much of Chapters 2–5 was on particular vector spaces and on particular vectors in those spaces. Most of those spaces were metric, normed, or inner product spaces. Now we will concentrate on classes of operators that transform vectors in one space into vectors in another space.
Christopher Heil

Chapter 7. Operators on Hilbert Spaces

In this chapter we will study operators that map one Hilbert space into another. The fact that we now have an inner product to work with allows us to derive much more detailed results than we could for operators on generic normed spaces.
Christopher Heil

Backmatter

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