Suitable for graduate students, this book reviews basic functional analysis focusing on the fundamental notion of completeness, demonstrating how it lies at the core of our understanding of mathematics. The theory is introduced step by step using examples and exercises; applications to other branches of mathematics are discussed in depth.

Adam Bobrowski is a professor in the Department of Mathematics at Lublin University of Technology, Poland. He was awarded the Hugo Steinhaus Prize for his achievements in analyzing mathematical models of biological reality and has authored more than 70 scientific papers and six books. His works include 'Functional Analysis for Probability and Stochastic Processes' (2005), 'Convergence of One-Parameter Operator Semigroups' (2016), and 'Generators of Markov Chains' (2020).

Introduction; 1. Complete metric spaces; 2. Banach's principle; 3. Picard's theorem; 4. Banach spaces; 5. Renewal equation in the McKendrick-von Foerster model; 6. Riemann integral for vector-valued functions; 7. The Stone-Weierstrass theorem; 8. Norms do differ; 9. Hilbert spaces; 10. Complete orthonormal sequences; 11. Heat equation; 12. Completeness of the space of operators; 13. Working in L(X); 14. The Banach-Steinhaus theorem and strong convergence; 15. We go deeper, deeper we go (into the structure of complete spaces); 16. Semigroups of operators; Appendix. Two consequences of the Hahn-Banach theorem; References; Index.