CRS Analysis 1

CMI, Aug-Nov 2024, Upendra Kulkarni
Rudin (1976), Abbott (2015), Tao (2016), Bartle & Sherbert (2010), Amann & Escher (2005)

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Compiled Notes

The real and complex number systems

• Ordered sets
• Bounds
• LUB property
• Fields
• Ordered fields
• The real field
• The extended real number system
• The complex field
• Euclidean spaces

Basic Topology

• Finite, countable, and uncountable sets
• Cantor’s Theorem
• Metric spaces

Sequences and series

• Sequences and convergence
• Subsequences
• Cauchy sequences
• LimSup and LimInf

Lecture Notes

• LEC ANA1 10 ✅
  • Started chapter 4. Motivated and stated 4.1. Limit points. 2.20. 3.2d.
• LEC ANA1 11 ✅
  • Continuity (4.5, 4.6, 4.7, 4.12). The sequence criterion (4.2). Algebra of limits of functions (4.4).
• LEC ANA1 12 skip for now
  • NEED to FINISH THIS STUFF! forget it. not gonna happen.
• LEC ANA1 13 ✅
  • Motivated and defined open and closed sets from 2.18. Proved 4.8 halfway.
• LEC ANA1 14 ✅
  • Finished proof of 4.8. Examples of open/closed sets. Showed that being closed implies containing all your limit points (This is Rudin’s definition of being closed. We defined being closed as the complement being empty, which rudin proves in 2.23). 2.24, 2.25, 2.26.
• LEC ANA1 15 ✅
  • 4.9, 4.10, 4.11. 2.29, 2.28, 2.27, 2.30.
• LEC ANA1 16 ✅
  • Sequential/limit point compactness and their equivalence. Intrinsic property of compactness. Examples of compact sets: closed boxes in ${\mathbb{R}}^k$ are bounded. A few proofs using sequential/limit point compactness. Heine Borel.
• LEC ANA1 17 ✅
  • Open cover compactness, equivalence of all three versions of compactness. Proofs using OCC.
• LEC ANA1 18 ✅
  • More proofs using OCC. 2.36, 3.10 b, 2.42
• LEC ANA1 19 ✅ ← Midsem syllabus ends here
  • Infinite limits and limits at infinity, discontinuities, monotonic functions. 4.25, 4.26, 4.27, 4.28, 4.29, 4.30, 4.32, 4.33, 4.34
• LEC ANA1 20 ✅
  • Uniform continuity
• LEC ANA1 21 ✅
  • Topological spaces, homeomorphisms, 4.17
• LEC ANA1 22 ✅ ← Quiz 2 syllabus ends here
  • Connected sets, Intermediate value theorem, connected components, allied theorems.
• LEC ANA1 23 ✅ Differentiation!
  • Derivatives, Algebra of derivatives, chain rule
• LEC ANA1 24 ✅
  • Mean value theorems, properties of derivatives
• LEC ANA1 25 ✅
  • Taylor’s theorem, MVT analogue for vector valued functions
• LEC ANA1 26 ✅ Integration!
  • Riemann and Stieltjes integrals, partitions, 6.1 through 6.6
• LEC ANA1 27 ✅
  • 6.8 through 6.12
• LEC ANA1 28 ✅
  • Fundamental theorem of calculus, integration by parts
• LEC ANA1 29 ✅ ← Midsem 2 syllabus ends here
  • Integration of vector valued functions, rectifiable curves
• LEC ANA1 30 ✅
  • Series, convergence tests, the number $e$
• LEC ANA1 31 ✅
  • Root test, ratio test, power series
• LEC ANA1 32 ✅
  • Real analytic functions
• LEC ANA1 33 ✅
  • Uniform convergence
• LEC ANA1 34
• LEC ANA1 35
• LEC ANA1 36
• LEC ANA1 37
• LEC ANA1 38

Excalidraw

These notes are either in excalidraw (which cannot be rendered by Quartz) or unformatted. Refer Compiled Notes for the content form these lectures.

• ANA1_L1 ✅
• ANA1_L2 ✅
• ANA1_L3 ✅
• ANA1_L4 ✅
• ANA1_L5 ✅
• LEC ANA1 6 ✅
• LEC ANA1 7 ✅
• LEC ANA1 8 ✅
• LEC ANA1 9 ✅

Homework

• ANA1_HW1, Solutions
• ANA1_HW2
• ANA1_HW3
• ANA1_HW3
• ANA1_HW4, ANA1_HW4.5
• ANA1_HW5
• ANA1_HW6

Tests

• Quizzes
  • ANA1_Q0
  • ANA1_Q1
  • ANA1_Q2
• ANA1_MidSem
• ANA1_MidSem2
• ANA1_EndSem

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References

Abbott, S. (2015). Understanding Analysis. Springer New York. https://doi.org/10.1007/978-1-4939-2712-8

Amann, H., & Escher, J. (Eds.). (2005). Analysis I. Birkhäuser Verlag. https://doi.org/10.1007/b137107

Bartle, R. G., & Sherbert, D. R. (2010). Introduction to Real Analysis (4. ed). Wiley.

Rudin, W. (1976). Principles of Mathematical Analysis (3d ed). McGraw-Hill.

Tao, T. (2016). Analysis I (Vol. 37). Springer Singapore. https://doi.org/10.1007/978-981-10-1789-6