# Course Website

## Math 6301: Analysis I

**Lecture:** MATH 6301-100

**Room:** Morton Hall 318

**Time:** MWF 9:40am -- 10:35am

**Instructor:** Marcel Bischoff

**Office:** Morton Hall 521

**Office Phone:** (740-59)3-1261

**Office Hours:**
Mon & Wed: 10:45am-11:45am, Fri: 1:00pm-2:00pm

### Schedule

- Mon 08/26:
- Syllabus
- Sets, Mappings, Equivalence relations
- Wed 08/28:
- Axiom of Choice, Zorn's Lemma
- Real Numbers
- Suggested Problem: Section 1.1, Problem 1.
- Fri 08/30:
- Section 1.1. and 1.2
- Natural Numbers, Mathematical Induction, Archimedean Property, Integers, Rational Numbers
- $\sqrt 2$ is not rational, $\mathbb Q$ is dense in $\mathbb R$
- Suggested Problems: Section 1.1, 1.2
- Mon 09/02: Labor Day (no class)
- Wed 09/04:
- Pigeonhole Principle, Cantor–Schröder–Bernstein Theorem, Countable Sets, Uncountable Sets
- Suggested Problems: Section 1.3
- Fri 09/06:
- $(a,b)$ is uncountable, Topological Space, Metric Space Topology
- Mon 09/09:
- Hand in: Homework 01
- Homework 02 uploaded
- $G_\delta$ sets, $F_\sigma$ sets, $\sigma$-algebras Borel sets
- Every open set of $\mathbb R$ is a countable disjoint union of open intervals.
- Points of closure, Closure, Neighbourhoods, Compactness
- Wed 09/11:
- Heine-Borel Theorem The Nested Set Theorem,
- Convergent sequences are bounded, the Monotone Convergence Criterion for Real Sequences
- Suggested problems: Section 1.4
- Fri 09/13:
- Bolzano-Weierstrass Theorem, Cauchy sequence, Cauchy's convergence theorem, Linearity of the limit
- Quiz 01
- Mon 09/16:
- More on Cauchy sequences, Limit superior and limit inferior, Series
- Suggested problems: Section 1.5
- Wed 09/18:
- Hand in Homework 02
- Continuous Functions, Extreme Value Theorem
- Intermediate Value Theorem, Uniform Continuity, Heine-Cantor Theorem, Monotone Functions
- Suggested problems: Section 1.6
- Fri 09/20:
- Outer measure
- Suggested Problems: Section 2.2
- Mon 09/23:
- The $\sigma$-algebra of Lebesgue measurable sets
- Suggested Problems: Section 2.2
- Uploaded Homework 03
- Wed 09/25:
- Proof that intervals are measurable
- Lebesgue Measure
- Suggested Problems: Section 2.3
- Outer/inner approximation of Lebesgue measurable sets
- Fri 09/27:
- Outer/inner approximation of Lebesgue measurable sets
- Suggested Problems: Section 2.4
- Countable additivity of Lebesgue measure, Continuity of Lebesgue Measure
- Fri 12/13: 8:00am-10am: Final Exam

### Ressources

- Lecture Notes (ohio.edu login required)
- Homework and Quizzes (ohio.edu login required)
- Textbook
- Tao - Analysis I (free eBook access through Alden Library)
- Tao - Analysis II (free eBook access through Alden Library)

Last update: 2019-11-29 21:37:44.