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!

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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.

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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.

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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.