Functional Analysis

Entering Hilbert Space

This book presents basic elements of the theory of Hilbert spaces and operators on Hilbert spaces, culminating in a proof of the spectral theorem for compact, self-adjoint operators on separable Hilbert spaces. It exhibits a construction of the space of pth power Lebesgue integrable functions by a completion procedure with respect to a suitable norm in a space of continuous functions, including proofs of the basic inequalities of Hölder and Minkowski. The Lp-spaces thereby emerges in direct analogy with a construction of the real numbers from the rational numbers. This allows grasping the main ideas more rapidly. Other important Banach spaces arising from function spaces and sequence spaces are also treated.

In this second edition, I have expanded the material on normed vector spaces and their operators presented in Chapter 1 to include proofs of the Open Mapping Theorem, the Closed Graph Theorem and the Hahn–Banach Theorem.

The material on operators between normed vector spaces is further expanded in a new Chapter 6, which presents the basic elements of the theory of Fredholm operators on general Banach spaces, not only on Hilbert spaces. This requires that we develop the theory of dual operators between Banach spaces to replace the use of adjoint operators between Hilbert spaces.

With the addition of the new material on normed vector spaces and their operators, the book can serve as a general introduction to functional analysis viewed as a theory of infinite dimensional linear spaces and linear operators acting on them.

Contents:

• Basic Elements of Metric Topology
• New Types of Function Spaces
• Theory of Hilbert Spaces
• Operators on Hilbert Spaces
• Spectral Theory
• Fredholm Theory
• Exercises

Readership: Undergraduates in mathematical and physical sciences, and mechanical, electrical and electronic engineering.
Key Features:

• Offers a fast construction of the Lebesgue integrable functions. These mathematical objects are introduced as added limits for sequences of continuous functions and are thereby experienced by students as being no more mysterious than the real numbers constructed as limits of sequences of rational numbers. No compromises are made with respect to mathematical rigor
• Contains a comprehensive discussion of the question about existence of a Schauder basis in a Hilbert space and the relation of this question to the topological notion of separability
• Contains a complete treatment of the spectral theorem for compact, self-adjoint linear operators on separable Hilbert spaces
• Gives an introduction to Fredholm theory. Fredholm operators are of interest far beyond functional analysis. They also play a significant role in theoretical physics, differential geometry and topology with the famous Index Theorem proved by Michael Atiyah and Isadore Singer in 1963 as a highlight