====== Math 104: Introduction to Real Analysis (2021 Fall) ======
$$\gdef\Q{\mathbb{Q}}$$

Instructor: Peng Zhou \\
Email: pzhou.math@berkeley.edu \\
Office: Evans 931 \\
Office Hour: Monday 12:10-1pm, **updated** Wednesday 10:10-11am, Friday 10:10-11am

Lecture: MWF, 11:10am - 12:00. Etcheverry 3107. 

GSI: James Dix. Mondays 9am-6pm, Wednesday 12-2pm in Evans 1049.

Online Help:
  * Zoom chat channel: search for "Math 104 with Peng Zhou", then you will find the channel. I will answer question there. 
  * My zoom office: https://berkeley.zoom.us/j/97935304012  time by appointment. 


==== Textbooks ====
  * Elementary Analysis: The Theory of Calculus, by Kenneth A. Ross.   [[https://link.springer.com/book/10.1007/978-1-4614-6271-2 | springer link]] (UC login required).
  * Principles of Mathematical Analysis, by Walter Rudin 
  * Introduction to analysis, by Terry Tao. ([[https://link.springer.com/book/10.1007%2F978-981-10-1789-6 | springer link ]])
  * notes from 2021 spring [[math104-s21:start|previous version]]
==== Grading ====
20% homework; 2 midterms 20% + 20%; and final 40%.  If you didn't do well in one of the midterm, you have the option to drop it, and final will have a 60% weight. The lowest homework grades will be dropped.

Midterm date: 9/22 (Wed), <del>11/3</del> 11/10 (Wed). There will be no make-up midterms. 

Final Date: Mon, Dec 13 • 11:30A - 2:30P

Homework will be submitted via gradescope. Entry Code:YVZRDZ

===== Schedule =====
part 1: number system,  sequence and limit, series. \\
part 2: metric space and topology. continuity. \\
part 3: differentiation and integration. 

==== Week 1 ====
  * Aug 25: introduction. counter-examples. Tao, Ch 1.
  * Aug 27: Peano Axioms for natural numbers. (Tao Ch2). Inadequacy of $\Q$. Least upper bound (Rudin Ch 1, section 1)
  * Reading homework; Tao Ch 1 and 2. Rudin Ch 1. 
  * [[HW1]]: due next Tuesday (Aug 31) 6pm

==== Week 2 ====
  * Aug 30: More about sup. Definition of field. 
  * Sep 1: Cauchy sequence of rational numbers.(Tao 5.2)
  * Sep 3: Equivalent Cauchy sequences of rational numbers as real numbers. Operation and properties of real numbers. (Tao 5.3)
  * [[hw2-sol]]: Due next **Thursday** 6pm. (Due date changed)

==== Week 3 ====
Tao 5.3-5.5
  * Sep 8: arithmetic operation on $\R$.
  * Sep 10: order on $\R$, and least upper bound property of $\R$. 
  * [[HW3]]: due next Tuesday (Sep 14) 6pm

==== Week 4 ====
Tao Ch 6. Ross Ch 2.1 - 2.7. 
  * Sep 13: Sequences in $\R$. Convergent implies Cauchy. Arithmetic operation commute with limit. Bounded monotone sequences are convergent. limsup.
  * Sep 15: $\pm \infty$. Tao 6.4. Cauchy sequences are convergent. 
  * Sep 17: Finish Cauchy sequence is convergent. Limit points and subsequence. 
  * [[HW4]] Due next Tuesday 6pm
  * [[math104-f21:midterm1-review]]

==== Week 5 ====
  * Sep 20 Subsequences, Countable set, $\R$ is not countable. 
  * Sep 22 [[Midterm 1]]
  * Sep 24 Various results from Ross section 10-12.  (No office hour today)
  * [[HW5]] due next Thursday 6pm.

==== Week 6 ====
  * Sep 27 Ross section 12
  * Sep 29 Series Ross 14,15. Root and Ratio test. 
  * Oct 1 finishing series, integral test. Start Metric space and topology. 
  * [[HW6]], Due next **Thursday** 6pm. (All future homeworks will be due on Thu 6pm)

==== Week 7 ====
  * Oct 4 Open sets in metric spaces. 
  * Oct 6 Examples of Metric spaces and topology. Metric on Graph. Metrics on $\R^2$, $l^1, l^2, l^p,  l^\infty$ metric. 
  * Oct 8 Limit points and closure. 
  * [[HW7]], Due next Thursday 6pm

==== Week 8 ====
  * Oct 11 Closure and Interior. Open covers and Compact sets
  * Oct 13 Compact sets are closed. Closed subset of compact set is compact. Compactness is absolute notion. (Rudin 2.30, 2.33, 2.34, 2.35)
  * Oct 15 Towards Thm 2.41. Finishing compactness. (will not talk about perfect set). [[math104-f21:compactness|sequential compactness and compactness]] 
  * [[HW8]] Due next Thursday 6pm. 


==== Week 9 ====
Wrapping up Ch 2. Continuity. Rudin Ch 4. Another concise lecture note to follow is Rui Wang's lecture note https://math.berkeley.edu/~ruiwang/pdf/104.pdf

**updated office hour from now on** <del>Tuesday 11-12am</del> moved to Wednesday 10:10-11am
  * Oct 18: Wrapping up loose ends in Ch 2: connected set. Sequential compactness and compactness. More examples.   
  * Oct 20: Begin Rudin Ch 4. Two definitions of continuous functions, using $\epsilon-\delta$, and use open sets.  
  * Oct 22: Example of Continuous functions. Do pre-image and image of continuous functions preserve open / closed / bounded / compact sets? 
  * [[HW9]] Due next Thursday 6pm


==== Week 10 ====
Continuity. 
  * Oct 25: Connectedness and Continuity. 
  * Oct 27: Operations on continuous function. (Cartesian product, composition, restriction of domain and codomain)
  * Oct 29: Limit of a function and discontinuity. 
  * [[HW10]]: Due next Thursday 6pm. 


==== Week 11 ====
Midterm 2 postponed to next Wednesday. 
  * Nov 1: Monotonic Functions (Rudin p95-98)
  * Nov 3: Uniform Continuity (Rudin p90-91), Intermediate Value Thm (Rudin Thm 4.23)
  * Nov 5: [[midterm2-review|Review for Midterm 2]]
  * [[HW11]] Due next Friday 6pm.
 
==== Week 12 ====
Sequences of functions. (Rudin Ch 7)

Office hour on Monday moved to Tuesday 12-2pm
  * Nov 8: Pointwise convergence and Uniform Convergence.
  * Nov 10: [[Midterm 2]] policy about cheat sheet is the same as midterm 1
  * Nov 12: Uniform convergence preserves continuity. Examples.
  * No HW this week 

==== Week 13 ====
Rudin Ch 5, Differentiation. 
One can also see notes from 2021 spring [[math104-s21:start|previous version]]

  * Nov 15: definition. examples. Chain rule. 
  * Nov 17: mean value theorem. 
  * Nov 19: L'hopital rule. Smooth Functions. Taylor theorem. 
  * [[HW13]]. Due Nov 29 Monday. 

==== Week 14 ====
Rudin Ch6

  * Nov 22 A brief encounter with Lebesbue measure theory and integration (optional). The definition of Riemann integrable functions. 
  * Nov 24 No class. 
  * Nov 26 No class

==== Week 15 ====
  * Office hour of GSI changed this week:  3pm - 6pm Tuesday and 9am-4pm Wednesday.
  * Videos from [[math104-s21:start|past semester]] are available on bcourse media gallery. You can use them for review.

  * Nov 29: Continuous function and Monotone functions are Riemann integrable. 
  * Dec 1: Riemann Stieltjes integral
  * Dec 3: Fundamental Theorem of Calculus 
  * [[HW15]], this is only for practice, not due.

==== Week 16 ====
Review week. No class. We have daily office hours 12-1pm, at [[https://berkeley.zoom.us/j/97935304012|zoom link]], from Monday-Thursday. If you plan to come, please arrive by 12:10. 

  * Tuesday office hour will be held in-person only. 12:10noon-1pm. 


==== Final ====

{{ math104-f21:math_104-final.pdf |final exam and solution}}, [[math104-f21:final-mistakes]], [[final-grades]]