Theoretical / proof-based
Graduate analysis · Graduate · Math
Learning objectives from the Graduate analysis syllabus, grouped by unit. Click an objective for study materials.
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)
Learning objectives
Click an objective for study materials.
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)
Definitions and structure
- 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
Proofs and reasoning
- Product measures and Fubini–Tonelli — Product measures and Fubini–Tonelli
- Normed spaces and completeness — Normed spaces and completeness
- Banach spaces — Banach spaces
- Linear operators — Linear operators
- Operator norm — Operator norm
Abstraction and generalization
- Hilbert spaces and orthogonality — Hilbert spaces and orthogonality
- Riesz representation — Riesz representation
- Hahn–Banach — Hahn–Banach
- Open mapping theorems (introduction) — Open mapping theorems (introduction)
- Weak and weak-* topologies (introduction) — Weak and weak-* topologies (introduction)
Modeling and computation
- Apply sigma-algebras, measurable functions, and measures in engineeri... — Apply sigma-algebras, measurable functions, and measures in engineering contexts
- Apply lebesgue integral in engineering contexts — Apply lebesgue integral in engineering contexts
- Apply convergence theorems (mct, fatou, dct) in engineering contexts — Apply convergence theorems (mct, fatou, dct) in engineering contexts
- Apply l^p spaces in engineering contexts — Apply l^p spaces in engineering contexts
- Apply hölder / minkowski inequalities in engineering contexts — Apply hölder / minkowski inequalities in engineering contexts
Data and technology
- Use software to explore graduate analysis problems numerically — Use software to explore graduate analysis problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Product measures and Fubini–Tonelli — Product measures and Fubini–Tonelli
- Normed spaces and completeness — Normed spaces and completeness
- Banach spaces — Banach spaces
- Linear operators — Linear operators
- Operator norm — Operator norm
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
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
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