**2017-1**

Advanced Calculus

Advanced Calculus

Instructor: Prof. Hyunseok Kim and Young-Ran Lee

TA: Hyunwoo Will Kwon

Office: R1416

Contact: willkwon [at] sogang [dot] ac [dot] kr

Office Hour: F 10:00-12:00 **(If these are not working for you, feel free to go Math Learning Center or make appointment by email)**

**TA Notes (Written in Korean)**

Week 1. Mathematical induction and well-ordering principle

Week 2. Completeness Axiom and Real Numbers

Week 3. Real numbers and limit of sequences

Week 4. Bolzano-Weierstrass theorem, Monotone convergence theorem and Cauchy sequence

Week 5. Limit of sequences, Limit of function

Week 6. Uniform continuity I

Week 7. Uniform continuity II, Monotone function

Week 8. Uniform continuity III, Monotone function

Week 9. Continuity: convexity, monotone function

Week 10. Differentiation

Week 11. Integration I

Week 12. Integration II

**Useful Links**

Teaching at the university level

Mathematical Writings

https://www1.maths.leeds.ac.uk/~khouston/pdf/htwm.pdf

Terry Tao’s advices

https://terrytao.wordpress.com/career-advice/

Writing the Greek Alphabet (Video)

**Supplementary**** Readings / Selected Homework Problems Explained**

Transcendency of Euler number $e$, irrationality of $\pi$

Counterexamples in differentiation

Weyl’s equidistribution theorem

**2017-1**

Introduction to Hilbert spaces (Sogang Math Honors Club)

Introduction to Hilbert spaces (Sogang Math Honors Club)

Coordinator: Hyunwoo Will Kwon

Textbook: N. Young, Introduction to Hilbert spaces, Cambridge University Press

This class is undergraduate mathematics seminar. We read some textbook related to our main topic. All students should give a presentation based on our main topic. The aim of this course is to understand the theory of Hilbert space and its application to various fields of mathematics.

As a coordinator, I do not give a presentation, but I will give some several comments during this seminar.

**Topics which are covered**

The following topics were considered in this seminar.

- Inner product spaces (Definition, Gram-Schmidt process)
- Brief introduction to the theory of Lebesgue integral (I gave a Lecture)
- Theory of Hilbert spaces (Riesz representation theorem, best approximation lemma, Parseval’s identity)

As an application of Hilbert space theory, we study some theory of Fourier series following Stein and Sharkarchi’s book.

- Basic properties of Fourier coefficients
- Convergence of Fourier series (Cesaro summerable, Abel summerable, $L^2$-convergence)
- Application to PDEs on unit disk
- Isoperimetric inequality
- Weyl’s equidistribution theorem.

Finally, we study spectral theory on Hilbert space. We follow Reed and Simon’s book, functional analysis. The following topics are considered.

- Bounded operators on Hilbert space
- Compact operators on Hilbert space
- Spectral decomposition of compact operators
- Spectrum and Resolvent of bounded operators
- Positive operators and the polar decomposition (if time permits)
- Trace class and Hilbert-Schmidt ideals
- The continuous functional calculus
- The spectral measures
- Spectral projections

**References**

- H. Brezis, Functional Analysis, Sobolev spaces and Partial Differential Equations
- L. C. Evans, Partial Differential Equation
- Gilbarg and Trudinger, Elliptic Partial Differential Equation
- P. Lax, Linear algebra and its application
- M. Reed and B. Simon, The methods of mathematical physics I: Functional Analysis
- E. M. Stein and R. Sharkarchi, Fourier Analysis, Real analysis
- Spence, Insel and Friedberg, Linear Algebra
- Wheeden and Zygmund, Measure and Integral
- K. Yosida, Functional Analysis