Attribute | Rank |
---|---|

Difficulty | 4 |

Proof Quality | 2 |

Readability | 3 |

Self-Study | 2 |

Good for Teaching? | 2 |

Quality of Exercises | 2 |

- Measure Theory
- rings and algebras of sets
- inner and outer measure
- completion of measures
- Lebesgue and Lebesgue-Stieltjes measures
- metric outer measure
- signed measures

- Integration
- measurable functions
- Egoroff's Theorem
- convergence in measure
- integrals of simple functions
- definition and properties of integrals
- sequences of integrable functions
- Lebesgue's Bounded Convergence Theorem
- the Riemann integral
- The Radon-Nikodym Theorem
- Lebesgue decomposition
- product of measures
- Fubini's Theorem

- Metric Spaces
*L*spaces^{p}- completeness
- compactness
- continuous functions on metric spaces
- The Stone-Weierstrass Theorem
- fixed-points

- Banach Spaces
- normed linear spaces
- subspaces and bases
- linear transformations
- The Principle of Uniform Boundedness
- The Open-Mapping and Closed-Graph Theorems
- The Hahn-Banach Theorem
- conjugate and reflexive spaces
- Tychonoff's Theorem
- weak topology in conjugate spaces
- adjoint operators

- Completely Continuous Operators
- Friedholm-Riesz-Schauder Theory
- spectral theory
- The Dirichlet Problem

- Hilbert Spaces
- The Projection Theorem
- projection operators
- orthonormal sets
- self-adjoint operators and the resolvent
- positive operators
- spectral families
- eigenvalue problems