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.

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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.

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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.