**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.

Joel TecsonHello,

Thanks for the review.

I had a question regarding what you got out of this course. While I don’t need it for my degree, I’m interested in taking Math 320. However, I’m hesitant due to its reputation for being very difficult.

Do you feel as if the content is well-geared towards strengthening your overall mathematical and logical reasoning? I’m actually an engineering student, so if you feel as if there’s merit towards engineering fields that would be very interesting as well.

arman rainaPost authorHi Joel!

Thanks for swinging by my blog.

The difficulty really depends on how much you got from prior proofy math courses (120,121,223,220,226,227,322,323 etc), what grades you find acceptable (you could check PAIR to see the kind of grades people got the last term if you haven’t already) and the amount of time you have. There were quite a few non-math people taking the course, and I think most found it quite hard but survived.

There are very few mentions of direct applications of the topics taught in the course to actual engineering or physics etc. However, if you do make the connections on your own you’ll find those analysis concepts do occur often behind the scenes when issues of convergence, smoothness and approximation come up. That said, I would guess you don’t generally need to worry too much about the proofs behind them working as an engineer. However, if you are interested in an advanced (research) degree relating to differential equations, statistics or economics etc analysis is supposed to be quite useful. Also, I believe any proofy math course would help general mathematical reasoning skills, and this is the most proof-based course I have taken so far. It also touches on a lot of fields in pure math (calculus, a bit of abstract algebra, topology and sets) so it is beneficial in that way as well.

Hope that helps 🙂

Joel TecsonHi Arman,

That’s super helpful. I think I will actually try to take it sometime before graduating. While I am studying engineering, I’d probably say that the course I’ve most enjoyed so far was math 220, so 320 seems like a good way of exploring more similar concepts.