MATH 130 P Mathematical Analysis


Schedule: 01:30 PM - 03:00 PM, Wed and Fri, KA 402
Credit: 3 units
Prerequisite: Math 55 (Elementary Analysis III)
Consultation: 02:00 PM - 05:00 PM Tue and Thu; 03:00 PM - 05:00 PM Wed and Fri (College of Science Dean's Office).
Course Outline:
     Link to Course Syllabus
     Link to Tentative Schedule of Lectures
  1. Sequences
    Metric Spaces, Convergence, Bounded Sets, Uniqueness of Limits, Subsequences, Real Sequences, Null Sequences, Elementary Rules, Comparison Test, Euclidean Spaces, Convergence in Product Spaces, Bounded Monotone Sequences, Infinite Limits, Limit Superior, Limit Inferior, Cauchy Sequences, Completeness
  2. Series
    Convergence of Series, Harmonic & Geometric Series, Calculation of Series, Alternating Series, Absolute Convergence, Rearrangement of Series, Double Series
  3. Continuity
    Continuity, Elementary Properties and Examples, Sequential Continuity, Algebra of Continuous Functions, Composition, One-sided Continuity, Open Sets, Closed Sets, Closure, Interior, Exterior, Boundary, Hausdorff Condition, Characterization of Continuous Functions, Continuous Extensions, Covers, Characterization of Compact sets, Sequential Compactness, Continuous Functions on Compact Sets, Extreme Value Theorem, Uniform Continuity, Connectivity, Intermediate Value Theorem
  4. Differentiability
    Derivative, Linear Approximation, Rules of differentiation, Chain Rule, Inverse functions, Differentiable Functions, Higher Derivatives, One-sided Differentiability, Partial Derivatives, Jacobian, Differentiability Criterion, Differentiation Rules, Mean-Value Theorems
  5. Integrability
    Riemann and Darboux Integrals, Properties of Darboux Integrals, Riemann Integrability
Reference Textbooks: Herbert Amann and Joachim Escher, Analysis I and Analysis II, Springer Basel, 2005 and 2008.

Exercises (deadline of submission in parenthesis): Link to Practice Problems

Schedule of Examinations:

MATH 232 SAT2 Real Analysis


Schedule: 09:00 AM - 12:00 NN, Sat, IB 103
Credit: 3 units
Prerequisite: COI (Consent of Instructor)
Consultation: 02:30 PM - 05:00 PM Mon, Tue, Wed, Thu, and Fri, or by Appointment (College of Science Dean's Office).
Course Outline (Link to Course Syllabus):
  1. Collection of Sets
    Rings, Algebras, Borel σ-algebras, Semirings, Dynkin Systems, Monotone Classes
  2. Measure Theory: Construction, Completion, and Examples
    Contents, Premeasures, Measures, From Premeasures to Measures, Lebesgue Measure, Outer and Inner Measures, Complete Measure Spaces, Borel and Lebesgue-Stieltjes Spaces, Regular Borel Measures on Metric Spaces, Image Measures, Properties of Lebesgue Measure
  3. Lebesgue Integration Theory and Convergence Theorems
    Measurable Real-Valued Functions, Lebesgue Integral of Simple Functions, Lebesgue Integral of Nonnegative Measurable Functions, Lebesgue Integrable Functions, Almost Everywhere Properties, Convergence Theorems, Riemann and Lebesgue Integrals
  4. Product Measures, Iterated Integrals, and Change of Variables Formula
    Initial, Final, and Product σ-algebras, Product Measures, Fubini–Tonelli Theorem, Integration Through Image Measures, Change of Variables for Integration
  5. Boundary Integration, Lebesgue Spaces, and Integral Inequalities
    Boundary Integrals, Gauss Divergence Theorem, Reynolds Transport Theorem, Lebesgue Spaces, Integral Inequalities: Jensen, Hölder, Minkowski, Lyapunov
  6. Decomposition of Measures
    Signed and Complex Measures, Radon–Nikodym Theorem, Lebesgue and Hahn Decompositions
Exercises (deadline of submission in parenthesis):