Book:Frigyes Riesz/Functional Analysis

Subject Matter

 * Functional Analysis

Contents
Part I: Modern theories of differentiation and integration


 * Chapter I: Differentiation


 * Lebesgue's Theorem on the Derivative of a Monotonic Function


 * 1. Example of a Nondifferentiable Continuous Function


 * 2. Lebesgue's Theorem on the Differentiation of a Monotonic Function. Sets of Measure Zero


 * 3. Proof of Lebesgue's Theorem


 * 4. Functions of Bounded Variation


 * Some Immediate Consequences of Lebesgue's Theorem


 * 5. Fubini's Theorem on the Differentiation of Series with Monotonic Terms


 * 6. Density Points of Linear Sets


 * 7. Saltus Functions


 * 8. Arbitrary Functions of Bounded Variation


 * 9. The Denjoy-Young Saks Theorem on the Derived Numbers of Arbitrary Functions


 * Interval Functions


 * 10. Preliminaries


 * 11. First Fundamental Theorem


 * 12. Second Fundamental Theorem


 * 13. The Darboux Integrals and the Riemann Integral


 * 14. Darboux's Theorem


 * 15. Functions of Bounded Variation and Rectification of Curves


 * Chapter II: The Lebesgue integral


 * Definition and Fundamental Properties


 * 16. The Integral for Step Functions. Two Lemmas


 * 17. The Integral for Summable Functions


 * 18. Term-by-Term Integration of an Increasing Sequence (Beppo Levi's Theorem)


 * 19. Term-by-Term Integration of a Majorized Sequence (Lebesgue's Theorem)


 * 20. Theorems Affirming the Integrbility of a Limit Function


 * 21. The Schwarsz, Holder and Minkowski Inequalities


 * 22. Measurable Sets and Measurable Functions


 * Chapter III: The Stieltjes integral and its generalizations


 * 23. The Total Variation and the Derivative of the Indefinite Integral


 * 24. Example of a Monotonic Continuous Function Whose Derivative Is Zero Almost Everywhere


 * 25. Absolutely Continuous Functions. Canonical Decomposition of Monotonic Functions


 * 26. Integration by Parts and Integration by Substitution


 * 27. The Integral as a Set Function


 * The Space $L^2$ and its Linear Functionals. $L^p$ Spaces


 * 28. The Space $L^2$; Convergence in the Mean; the Riesz-Fischer Theorem


 * 29. Weal Convergence


 * 30. Linear Functionals


 * 31. Sequence of Linear Functionals; a Theorem of Osgood


 * 32. Separability of $L^2$. The Theorem of Choice


 * 33. Orthonormal Systems


 * 34. Subspaces of $L^2$. The Decomposition Theorem


 * 35. Another Proof of the Theorem of Choice. Extension of Functionals


 * 36. The Space $L^p$ and Its Linear Functionals


 * 37. A Theorem on Mean Convergence


 * 38. A Theorem of Banach and Saks


 * Functions of Several Variables


 * 39. Definitions. Principle of Transition


 * 40. Successive Integrations. Fubini's Theorem


 * 41. The Derivative Over a Net of a Non-negative, Additive Rectange Function. Parallel Displacement of the Net


 * 42. Rectangle Functions of Bounded Variation. Conjugate Nets


 * 43. Additive Set Functions. Sets Measurable $\paren B$


 * Other Definitions of the Lebesgue Integral


 * 44. Sets Measurable $\paren L$


 * 45. Functions Measurable $\paren L$ and the Integral $\paren L$


 * 46. Other Definitions. Egoroff's Theorem


 * 47. Elementary Proof of the Theorems of Arzela and Osgood


 * 48. The Lebesgue Integral Considered as the Inverse Operation of Differentiation

'''Part II: Integral equations. Linear transforms'''


 * Chapter IV: Integral equations


 * Chapter V: Hilbert and Banach spaces


 * Chapter VI: Completely continuous symmetric transformations of Hilbert space


 * Chapter VII: Bounded symmetric, unitary, and normal transformations of Hilbert space


 * Chapter VIII: Unbounded linear transformations of Hilbert space


 * Chapter IX: Self-adjoint transformations. Functional calculus, spectrum, perturbations


 * Chapter X: Groups and semigroups of transformations


 * Chapter XI: Spectral theories for linear transformations of general type

Bibliography

Appendix

Index

Notation & symbols