Tag Archives: Real analysis

Course Review: MATH 418

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Probability

“I heard you had a lot of difficulty with the last homework. So you will be relieved to note that next homework you will have another opportunity to practice similar problems.”

Text:  A First Look at Rigorous Probability Theory, 2nd ed by J.S. Rosenthal

Prof: Dr Gordon Slade

Prof Slade is very clear and keeps things simple. He also has no problem slowing down to answer questions from students if they are not following along. He has a dry sense of humor, that keeps the class interesting, and he can even by funny when he is talking in earnest.

Difficulty

After taking measure theory, several sections of the course can feel like review. This was a good thing for me, as I found MATH 420 a tad fast. There are a handful of new techniques that you learn in the homework and class, but there are not that many new concepts if you have some background in measure theory and probability. MATH 421 material also comes up in terms of weak convergence.

Key Concepts

Probability Triples

Random Variables

Distributions

Expectation

Borel-Cantelli

Modes of Convergence

Law of Large Numbers

Central Limit Theorem

Characteristic Functions

Hard Concepts

Tail Events: Kind of funny to think about. Also, include definitions of limit supremum and limit infimum for sequences of event which can be difficult to convert to statements about limits of random variables.

Weak Convergence: There are a lot of equivalent statements, and if you pick the wrong one it can be a mission to prove that convergence occurs.

Conclusion

Good review of measure theory, and gives you a mathematical foundation to elementary probability.

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Course Review: MATH 421

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Real Analysis II

“If everyone is getting closer together, and you are going to Cleveland, then everyone’s going to Cleveland!”

Text: Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland

Prof: Dr Brian Marcus

Prof Marcus is one of the kindest professors I know. He also explains each concept from multiple perspectives and breaks the concepts down to simple intuition. His notes are super useful and he has a great sense of humor.

Difficulty

The classes moves somewhat slower then 420, so we went a lot more deeper into the details of certain proofs and where they come from. This meant the classes were easier to follow (and in my view more enjoyable!). Further the homework was largely straightforward. The one challenge in this course was the material towards the latter half of the term was slightly more complex and were not assessed on homework. We also may not have covered the material generally covered in a similar course.

Key Concepts

Banach spaces

Hilbert spaces

Linear functionals

Hahn Banach Theorem

Open mapping theorem

Uniform boundedness

Riesz representation theorems

Hard Concepts

Weak Convergence: Can be hard to think about neighborhoods in this topology.

Hilbert Spaces: There are a lot of nifty tricks in Hilbert spaces that are not immediately obvious, and do not work in other spaces.

Conclusion

Really fun course and learnt a lot. Nice conclusion to my studies of analysis in some sense.

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Course Review: MATH 420

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Real Analysis I

“F just comes along for the ride…”

Text: Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland

Prof: Dr Stephen Gustafson

Prof Gustafson is an engaging professor. He is able to give a quick, intuitive overview of quite complex topics in class, leaving us to work out the technical details on our own. He also is very helpful during office hours. He also posted his notes online.

Difficulty

The classes move very fast so one might find it challenging keeping up with the readings. Also, though the classes are easy to follow, the homework is increasingly challenging and time-consuming. The final exam was quite hard, practice and recalling theorems and definitions are all important. This course covers a lot of material, so watch out!

Key Concepts

Sigma-Algebras

Measures

Differentiation of Measures

Lebesgue integration

Convergence of Functions

LP spaces

Hard Concepts

Lebesgue Radon Nikodyn Theorem: Quite powerful theorem, need to notice where to apply it. Also can be hard to find the decomposition at times.

Fatou’s Lemma: Tricky working with the limit infimum, also quite powerful and pops up in unexpected places.

Conclusion

One of the harder courses I have ever taken. Started learning some really powerful theorems about convergence, for example.

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Course Review: MATH 321

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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.

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Course Review: MATH 320

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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.

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