Graduate analysis
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Analysis — scope drawn from Axler Measure, Integration & Real Analysis (open access), Bass Real Analysis for Graduate Students (free PDF), and typical first-year graduate analysis syllabi.
Measure and integration
- Sigma-algebras, measurable functions, and measures — Sigma-algebras, measurable functions, and measures
- Lebesgue integral — Lebesgue integral
- Convergence theorems (MCT, Fatou, DCT) — Convergence theorems (MCT, Fatou, DCT)
- L^p spaces — L^p spaces
- Hölder / Minkowski inequalities — Hölder / Minkowski inequalities
- Product measures and Fubini–Tonelli — Product measures and Fubini–Tonelli
Banach and Hilbert spaces
- Normed spaces and completeness — Normed spaces and completeness
- Banach spaces — Banach spaces
- Linear operators — Linear operators
- Operator norm — Operator norm
- Hilbert spaces and orthogonality — Hilbert spaces and orthogonality
- Riesz representation — Riesz representation
- Hahn–Banach — Hahn–Banach
- Open mapping theorems (introduction) — Open mapping theorems (introduction)
Functional analysis topics
- Weak and weak-* topologies (introduction) — Weak and weak-* topologies (introduction)
- Compact operators — Compact operators
- Fredholm theory (overview) — Fredholm theory (overview)
- Spectral theorem for bounded self-adjoint operators (introduction) — Spectral theorem for bounded self-adjoint operators (introduction)
Applications and problem sets
- Research-level problem sets aligned to your course syllabus — Research-level problem sets aligned to your course syllabus
- Connections to PDE — Connections to PDE
- Probability (as covered in your program) — Probability (as covered in your program)
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Measure and integration
Sigma-algebras, measurable functions, and measures
Coming soon - Banach and Hilbert spaces
Normed spaces and completeness
Coming soon - Functional analysis topics
Weak and weak-* topologies (introduction)
Coming soon - Applications and problem sets
Research-level problem sets aligned to your course syllabus
Coming soon
Notes
Graduate Analysis — scope drawn from Axler Measure, Integration & Real Analysis (open access), Bass Real Analysis for Graduate Students (free PDF), and typical first-year graduate analysis syllabi. Topic outline: `content/topics/graduate/analysis.json`.