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--- math104-s21:s:morganmakhina [2021/05/06 03:41]
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+++ math104-s21:s:morganmakhina [2026/02/21 14:41] (current)
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-====Morgan's Real Analysis Review Page====
+=====Morgan's Real Analysis Review Page=====
+====Number systems:====
-=== Some Questions===
+**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 $\R$ that other number systems don't have? And, by the way: what are some properties of $\N$ that we use in real analysis (perhaps sometimes taking them for granted)?
+**
-==Number systems, sequences, limits, &c==
-**1-4) What are real numbers, anyway? Why do we need them? How can we rigorously define (ie, construct) them? What are some properties of $\R$ that other number systems don't have?
-**
-**5) Sets, sequences, series... what's the difference?**
+====Sets, sequences, series...====
-**6)What's a Cauchy Sequence?**
+**6) What's the difference between sets, sequences, and series??**
-**7) How can you tell if a sequence converges (and, if it does, how can you tell what it converges to)? **
+**7)What's a Cauchy Sequence?**
-**8) How can you tell if a series converges (and, if it does, how can you tell what it converges to)? **
+**8) How can you tell if a sequence converges (and, if it does, how can you tell what it converges to)? **
-**9)What is radius of convergence?**
+**9) How can you tell if a series converges (and, if it does, how can you tell what it converges to)? **
-==Metric Spaces==
+**10)What is radius of convergence?**
-**What's a metric space?
+**11) Why do we care about monotone sequences?**
+====Topology:====
+**What's a metric space?**
 ** What are some familiar and less familiar metrics (distance functions)?**
@@ Line 27: / Line 30: @@
 ** 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's a complete metric space?**
-==Topology==
+**What are topological spaces, and how is this notion different from that of a 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?**
-Topological concepts are intuitive... until they're not. What are some caveats to watch out for?
-What are some of the particularly useful results in this section?
+**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?
+**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 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'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 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's a continuous function? What conclusions can we make if we know a function is continuous? What conclusions might we be tempted to make, that actually aren't true?
+**What is uniform continuity?**
+====Sequences of Functions:====
-What is uniform continuity?
+**What is the difference between pointwise and uniform convergence?**
-==Sequences of Functions==
+** 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?
+**
-What is the difference between pointwise and uniform convergence? What are some examples of sequences of functions that converge pointwise but not uniformly?
+==== Derivatives:====
-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 are some of the key theorems in this section?**
-What conclusions might we be tempted to make about uniformly converging sequences (of functions) that aren't actually true?
+**What are some surprising results in this section?**
+**When do Taylor series approximations fail?**
-== Derivatives, Integrals, etc==
+**What is Taylor's Theorem, and why is it useful?**
-What are some of the key theorems in this section?
-What are some surprising results in this section?
+====Integration:====
-When do Taylor series approximations fail?
-What is Taylor's Theorem, and why is it useful?
+====Extras:====
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-===Bonus Questions===
+====Bonus Questions:====
 **What were some of the particularly surprising, memorable, and fun things I learned in this course?**

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