Introduction to the Analysis of Normed Linear Spaces

1. Basic properties of normed linear spaces; 2. Classes of example spaces; 3. Orthonormal sets in inner product spaces; 4. Norming mappings and forming duals and operator algebras; 5. The shape of the dual; 6. The Hahn–Banach theorem; 7. The natural embedding and reflexivity; 8. Subreflexivity; 9. Baire category theory for metric spaces; 10. The open mapping and closed graph theorems; 11. The uniform boundedness theorem; 12. Conjugate mappings; 13. Adjoint operators on Hilbert space; 14. Projection operators; 15. Compact operators; 16. The spectrum; 17. The spectrum of a continuous linear operator; 18. The spectrum of a compact operator; 19. The spectral theorem for compact normal operators on Hilbert space; 20. The spectral theorem for compact operators on Hilbert space; Appendices. A1. Zorn's lemma; A2. Numerical equivalence; A3. Hamel basis.