Functional Analysis Revisited

Texte du rabat

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.

Contenu

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.