# Course Review: MATH 321

Real Variables II

“And now we have the tools the derive the Fundamental Theorem of Calculus, the formula you’ve been using since you were children.”

Text: Principles of Mathematical Analysis by Walter Rudin (3rd Edition)

Prof:  Dr. Gordon Slade

Dr. Slade was an engaging professor. He was clear while also motivating the content well. Once in a while, he would drop a wry joke, which brought a small amount of necessary levity to the class. One thing that stood out to me is that he would welcome clarifications and inquiries/corrections on the smallest details, especially on Piazza, rather than dismissing them which I think is helpful in such a class. Students’ often lack confidence in their first few serious proof courses and sometimes misunderstandings on seemingly small details can exacerbate that.

Difficulty

I found the course content more manageable than MATH 320, as it felt more well-motivated and we were applying a lot more of the theorems learnt in MATH 320. On the other hand, the homework was slightly harder as the proofs became more lengthy. Sometimes what was an entire proof question in MATH 320 is just a minor trick as part of proving some larger theorem. The midterms and final were all mostly doable, though I believe I made several trivial errors in the final. The final grades appeared to be scaled.

Key Concepts

Riemann-Stieltjes integral

Sequences of functions

Equicontinuity

Stone-Weierstrass Theorem

Arzela-Ascoli Theorem

Power Series

Fourier Series

Hard Concepts

Metric space of functions: Takes a while getting used to discussing metric spaces where points are functions. Need to understand what neighbourhoods and open and closed sets refer to in this context.  Then you can apply a lot of the theorems already learnt to this metric space.

Inequalities: It may seem ridiculous to mention this, but some of the hardest questions involve finding tight bounds on functions using simple tricks such as triangle inequality, Cauchy, tangent lines, sums of squares in clever ways.

Examples: Coming up with examples (e.g. a point or a sequence) is not only useful in questions that explicitly ask for them, it can also be applied in conjunction with certain theorems taught in the course to show a certain object does not have a certain property. It is also a useful thing to practice before an exam as it helps you remember the conditions for certain theorems.  They can be really tricky to come up with on the spot sometimes.

Conclusion

Rewarding continuation of Real Variables I. Content was well motivated and homework was challenging.

# Course Review: MATH 320

Real Variables I

“You don’t really need a metric. All you need is the open sets.”

Text: Principles of Mathematical Analysis by Walter Rudin (3rd Edition)

Prof:  Dr Joshua Zahl

Dr Zahl was a structured, clear professor. He also provided useful insights into the ideas behind various proofs. I don’t think he has taught this course many times before, so he is probably still in the process of fine-tuning his delivery. Also, his surname literally means ‘number’ in German!

Difficulty

While the readings are pretty dense, the homework was quite doable by honours mathematics standards. I found the first midterm surprisingly easy. The second midterm was significantly harder, and I had not fully understood the concept of compact sets, so I did not do well at all. The class also got wrecked so there was a lot of scaling. The final was very reasonable.

Key Concepts

Properties of Real Numbers

Metric Spaces

Open Sets

Sequences

Continuous Functions

The Derivative

Hard Concepts

Compact Sets: This concept has a deceptively Byzantine definition but is actually really fundamental to the course. Reading the history of the concept from Wikipedia and understanding many examples/counter-examples of sets that are or are not compact gave me a better intuition.

Sequences: Not a hard concept to understand, however an invaluable tool in certain seemingly intractable problems. Many times, it’s helpful to construct a sequence of points or even intervals and then use properties of such sequences in that space to prove the theorem.

Conclusion

Tough though doable class, if you have any background in mathematical proofs. One of the key learnings I took out this class, was the importance of spending time understanding complex definitions.