MATH447

MATH447 (Real Variables) is an introductory, three credit hour Real Analysis course targeted towards undergraduates with prior proofs exposure. It is more rigorous and in-depth than MATH444 (Elementary Real Analysis).

Content Covered

• Sequences (both in the reals and arbitrary metric spaces)
• Limits of sequences
• Continuity of functions (both in the reals and arbitrary metric spaces)
• Sequences of functions
• Introductory topology in metric spaces
• Differentiation
• Integration

Depending on the instructor, additional topics, such as formal development of exponentials and logarithms, may be included.

Prerequisites

• MATH241 (Calculus III)
• MATH347 (Fundamental Mathematics)

It is important to have a solid grasp of proof writing before entering MATH447. Students with this understanding who have not completed MATH347 (or MATH241) can enroll into the course with instructor consent.

When to Take It

MATH447 is most frequently taken as part of a math minor, and doesn't serve as a prerequisite for any undergraduate CS or ECE courses, so it can be taken at any time. The course is best taken while a student is still fresh with proof-writing, whether through MATH347 or other proof-heavy courses, such as CS374A (Intro to Algorithms and Models of Computation).

Course Structure

The structure of MATH447 can be instructor-dependent, but typically there are weekly homework assignments with between two and three midterms and one final exam. Homework assignments contain almost exclusively proof-based questions, and difficulty can vary based on instructor and even week-to-week. Expect to spend between three and eight hours on average.

MATH447 meets three times per week for 50 minutes each. Midterm exams are typically given during the course meeting, and thus typically contain relatively straightforward problems (but are still proof-based).

Instructors

MATH447 instructors change every semester, and are typically not announced until just before the beginning of the semester. It is typically taught by instructors whose research is at least somewhat related to analysis.

Course Tips

MATH447 formalizes many notions students are already familiar with from Calculus I-III. It is important to build a geometric/pictoral understanding of concepts discussed in the course, as this will help supplement intuition when coming up with ideas for proofs, especially related to more abstract concepts such as metric spaces.

Many analysis proofs can be technical, and require lots of "inequality bashing," that is, clever uses of inequalities to prove a statement. It may be helpful to keep note of all inequalities (triangle inequality, reverse triangle inequality, etc) covered in the course to keep as reference when completing homework assignments.

As always, office hours are a great resource, and are often quite empty for MATH447.

Life After

MATH447 is a prerequisite for MATH448 (Complex Variables), and prepares students for graduate coursework in mathematics and computer science (especially for the theory behind machine learning).

The proof-heavy nature of MATH447 will improve your proof-writing ability and help you in future proof-based courses.

Infamous Topics

• Topology of metric spaces: concepts such as open/closed and connected/disconnected can be confusing at first. It is helpful to first visualize these concepts when applied to the Reals, and then apply this intuition to the general case.
• Defining continuity and compactness of functions in metric spaces: without a solid understanding of the above, definitions will feel unintuitive.

Resources

We haven't found any external resources that are super useful for this course yet. If you have suggestions, feel free to open an issue on GitHub.