Analysis 1

• Profs: Ö. Imamoglu
• Website: https://metaphor.ethz.ch/x/2025/fs/401-0212-16L/
• SAM : https://sam-up.math.ethz.ch/
• Admin: Admin
• Material: Material
• Videos:
  • 2024: Videos ETHZ
  • 2025: Videos ETHZ

Lecture Notes

• 01 Introduction, Real Numbers, Completeness
• 02 Real Numbers, Min, Max, Bounds, Supremum, Infimum, Cardinality, Euclidian Space
• 03 Supremum, Infimum, Cross Product, Complex Numbers, Sequences
• 04 Divergence, Limits, Monotony, Weierstrass Theorem, Limit Superior and Inferior
• 05 Sandwich Lemma, Cauchy Criterion, and Bolzano-Weierstrass Theorem
• 06 Accumulation Points, Sequences in Reals and Complex, and Series
• 07 Series Cauchy Criterion, Absolute and Conditional Convergence, Riemann Rearrangement Theorem, Dirichlet’s Theorem
• 08 More Convergence Criteria, Dirichlet’s Theorem, Ratio, Root Tests and Power Series
• 09 Double Series, Cauchy Products, Exponential Functions and Real-Valued Functions
• 10 Continuity, Intermediate Value Theorem
• 11 Continuity, Extreme Value Theorem, Composition of Continuous Functions, Continuity of Inverse Functions
• 12 Continuity, Exponential Function and Function Sequences
• 13 Function Sequences, Continuity of Limit Function under Uniform Convergence, Cauchy Criterion for Uniform Convergence, Series of Functions, Uniform Convergence of Power Series, Trig Functions

Script

Note that this is WIP and might include inaccuracies and mistakes… (Ch4-5 aren’t published yet…)

• Chapter 1 - Real Numbers, Euclidean Spaces, and Complex Numbers
• Chapter 2 - Sequences and Series, Approaching Infinity, Divergence, Infinity, Algebra of Limits, Monotone Sequences, Weierstrass, Cauchy, Series, Summing Infinitely, Special Series and Operations
• Chapter 3 - Continuous Functions, Smoothness and Limits, Continuity, Functions Without Jumps, Combining Continuous Functions, Fundamental Theorems of Continuity, Exponential and Trigonometric Functions