ANALYSIS II

Prerequisites & course information

Requisites: MATH 521 and ( MATH 320 , 340 , 341 , 345 , or 375 ), graduate/professional standing, or declared in Mathematics VISP (graduate or undergraduate) Course Designation: Core GenEd - Natural Science & Wellness Breadth - Natural Science Level - Advanced L&S Credit - L&S Liberal Arts and Science Grad 50% - Counts toward 50% graduate coursework requirement Repeatable for Credit: No Last Taught: Fall 2026 Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in advanced real analysis (e.g., uniform and pointwise convergences, Fourier series, compactness, metric spaces, etc.). Audience: Both Grad & Undergrad 2. Use the above definitions to prove if a mathematical construct does or does not have the condition of being a particular mathematical object or having a particular property (e.g. that a sequence of functions converges uniformly or not, whether a set in a metric space is compact or not, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of advanced real analysis. (e.g., Arzela-Ascoli theorem, Stone-Weierstrass theorem, the Contraction Principle etc.), and recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Use the above theorems in the context of longer arguments by examining their premises (e.g., using the Arzela-Ascoli theorem to check if a set of continuous functions is relatively compact or not). Audience: Both Grad & Undergrad 5. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples, etc.). Audience: Both Grad & Undergrad 6. Convey arguments in oral and written forms using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 7. Identify applications of course content in current areas of research. Audience: Graduate Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in advanced real analysis (e.g., uniform and pointwise convergences, Fourier series, compactness, metric spaces, etc.). Audience: Both Grad & Undergrad 2. Use the above definitions to prove if a mathematical construct does or does not have the condition of being a particular mathematical object or having a particular property (e.g. that a sequence of functions converges uniformly or not, whether a set in a metric space is compact or not, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of advanced real analysis. (e.g., Arzela-Ascoli theorem, Stone-Weierstrass theorem, the Contraction Principle etc.), and recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Use the above theorems in the context of longer arguments by examining their premises (e.g., using the Arzela-Ascoli theorem to check if a set of continuous functions is relatively compact or not). Audience: Both Grad & Undergrad 5. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples, etc.). Audience: Both Grad & Undergrad 6. Convey arguments in oral and written forms using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 7. Identify applications of course content in current areas of research. Audience: Graduate