User Tools

Site Tools


math104:start

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. springer link (UC login required).
  • Principles of Mathematical Analysis, by Walter Rudin
  • Introduction to analysis, by Terry Tao. ( springer link )

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), 11/3 (Wed). There will be no make-up midterms.

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.
  • HW 1 (with solution): 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)
  • HW 2 (with Solution): 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 $\R$.
  • HW 3 with solution: 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.
  • HW 4 Due next Tuesday 6pm

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)
  • HW 5 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.
  • HW 6, 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.
  • HW 7, 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). sequential compactness and compactness
  • HW 8 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 Tuesday 11-12am 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?
  • HW 9 Due next Thursday 6pm

Week 10

Continuity.

  • Oct 25: Connectedness and Continuity.
  • Oct 27: Limit of a function and discontinuity.
  • Oct 29: Monotone function.

Week 11

Midterm 2 postponed to next Wednesday.

math104/start.txt · Last modified: 2021/10/26 22:18 by pzhou