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math104-s21:s:morganmakhina [2021/05/04 22:58]
107.77.212.37 		math104-s21:s:morganmakhina [2022/01/11 10:57] (current)
pzhou ↷ Page moved from math104-2021sp:s:morganmakhina to math104-s21:s:morganmakhina
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-====Morgan's Real Analysis Review Page====+=====Morgan's Real Analysis Review Page=====
  
 +====Number systems:====
 +
 +**1-5) What are real numbers, anyway? Why do we need them? How can we rigorously define (ie, construct) them? What are some properties of R\R that other number systems don't have? And, by the way: what are some properties of N\N that we use in real analysis (perhaps sometimes taking them for granted)?
 +**
 +
 +
 +
 +====Sets, sequences, series...====
 +
 +**6) What's the difference between sets, sequences, and series??**
 +
 +**7)What's a Cauchy Sequence?**
 +
 +**8) How can you tell if a sequence converges (and, if it does, how can you tell what it converges to)? **
 +
 +**9) How can you tell if a series converges (and, if it does, how can you tell what it converges to)? **
 +
 +**10)What is radius of convergence?**
 +
 +**11) Why do we care about monotone sequences?**
 +
 +====Topology:====
 +
 +**What's a metric space?**
 +
 +** What are some familiar and less familiar metrics (distance functions)?**
 +
 +** What are some examples of functions that aren't distance functions, even though they have some properties in common with distance functions?**
 +
 +**What's a complete metric space?**
 +
 +**What are topological spaces, and how is this notion different from that of a metric space?**
 +
 +**Topological concepts are intuitive... until they're not. What are some caveats to watch out for?**
 +
 +
 +**Which properties of topological subspaces depend on the ambient space, and which do not?**
 +
 +**Compact vs closed & bounded: when are these equivalent? When are they not equivalent?**
 +
 +**What does "sequentially compact" mean, and when is this property equivalent to compactness?**
 +
 +
 +**What's so special about compact sets? (Ie, what are some theorems we proved about compact sets that won't hold for other kinds of sets?)**
 +
 +**What is the Heine-Borel Theorem? When can we apply it, and when should we not apply it?**
 +
 +**What is the Bolzano-Weierstrass Theorem, and how does it relate to the Heine-Borel Theorem?**
 +
 +**What are some of the particularly useful results in this section?**
 +
 +====Continuity:====
 +
 +**What's a continuous function?**
 +
 +** What conclusions can we make if we know a function is continuous?**
 +
 +** What conclusions might we be tempted to make about continuous functions that actually aren't true?
 +**
 +
 +**What is uniform continuity?**
 +
 +====Sequences of Functions:====
 +
 +**What is the difference between pointwise and uniform convergence?**
 +
 +** What are some examples of sequences of functions that converge pointwise but not uniformly?**
 +
 +**What conclusions can we make about uniformly converging sequences of functions that would no longer necessarily be valid if we replaced uniform convergence by pointwise convergence?**
 +
 +**What conclusions might we be tempted to make about uniformly converging sequences (of functions) that aren't actually true?
 +**
 +
 +==== Derivatives:====
 +
 +**What are some of the key theorems in this section?**
 +
 +**What are some surprising results in this section?**
 +
 +**When do Taylor series approximations fail?**
 +
 +**What is Taylor's Theorem, and why is it useful?**
 +
 +
 +====Integration:====
 +
 +
 +
 +
 +====Extras:====
 +
 +
 +Note: The following questions appeared on Anton's review page (which I found very useful!). I found them too important to omit, and too well-stated to paraphrase. The two exam-related questions were ones that I did not get right on the midterm. 
 +
 +** What is the Weierstrass M-Test?**
 +
 +** Why is the set [0,1]∩Q[0,1] \cap \mathbb{Q} not compact while [0,1][0,1] is? (MT2, Q1, (4))**
 +
 +** Why is the set {0}∪{1/n∣n∈N}\{0\} \cup \{1/n | n \in \mathbb{N}\} compact? (MT2, Q1, (5))**
 +
 +
 +====Bonus Questions:====
 +
 +**What were some of the particularly surprising, memorable, and fun things I learned in this course?**
 +
 +**Briefly list the most significant concepts/theorems covered in this course.**
 +
 +**Where can I find some sample exams to do for practice?**
  
  
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 (This is a work in progress, and organization will improve soon!) (This is a work in progress, and organization will improve soon!)
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 1) Number systems: N\N, Z\Z, Q\mathbb Q, R\R, C\C, others, & some of their properties 1) Number systems: N\N, Z\Z, Q\mathbb Q, R\R, C\C, others, & some of their properties
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 Every convergent sequence has a monotone subsequence Every convergent sequence has a monotone subsequence
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math104-s21/s/morganmakhina.1620194297.txt.gz · Last modified: 2021/05/04 22:58 by 107.77.212.37

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