This textbook provides an introduction to the methods and language of functional analysis, including Hilbert spaces, Fredholm theory for compact operators, and spectral theory of self-adjoint operators. It also presents the basic theorems and methods of abstract functional analysis and a few applications of these methods to Banach algebras and the theory of unbounded self-adjoint operators.

The text corresponds to material for two semester courses (Part I and Part II, respectively) and is essentially self-contained. Prerequisites for the first part are minimal amounts of linear algebra and calculus. For the second part, some knowledge of topology and measure theory is recommended. Each of the 11 chapters is followed by numerous exercises, with solutions given at the end of the book.

The text is ideal for a one-year course. It will also provide a sound basis for further study. It is suitable for graduate students and researchers interested in operator theory and functional analysis.

Yuli Eidelman and Vitali Milman, Tel Aviv University, Israel, and Antonis Tsolomitis, University of the Aegean, Samos, Greece.

1. Linear Spaces; normed spaces; first examples
2. Hilbert Spaces
3. The dual space
4. Bounded linear operators
5. Spectrum. Fredholm theory of compact operations
6. Self-adjoint operators
7. Functions of operators; spectral decompositions

1. Spectral theory of unitary operators
2. The Fundamental theorems and the basic methods
3. Banach algebras
4. Unbounded self-adjoint and symmetric operators in H