## Mathematical Analysis III, NMAI056, winter term 2019/20

Lectures are on Thursdays in 15:40 - 17:10 in the lecture room S1 at the 4th floor in the Malostranske namesti building.
I am also teaching tutorial to it (Thursday 17:20-18:50 in S1), see the
tutorial page for conditions under which you can get a ''zapocet''. Concerning exam, it will be possible to register for it in the SIS
information system and the exam questions are here.

1st lecture on October 3, 2019. Chapter 1: Metric spaces. Examples of metric spaces. Lecture notes for the 1st lecture (updated on October 22).

2nd lecture on October 10, 2019. Normed fields, Ostrowski's theorem, review. Lecture notes for the 2nd lecture (updated on October 22).

3rd lecture on October 17, 2019. Compactness, the Heine-Borel theorem, relation of a point and a subset, homeomorphisms.
Lecture notes for the 3rd lecture.

4th lecture on October 24, 2019. Homeomorphisms, handcuffs, connected spaces, complete spaces.
Lecture notes for the 4th lecture.

5th lecture on October 31, 2019. The Baire theorem, perfect sets.
Chapter 2: Sequences and series of functions. Types of convergence of functions.
Lecture notes for the 5th lecture.

6th lecture on November 7, 2019. Uniform convergence and normed spaces of functions,
complete spaces of functions, a continuous function without derivative.
Lecture notes for the 6th lecture .

7th lecture on November 14, 2019. Proof of existence of a continuous but non-differentiable
function. Lecture notes for the 7th lecture.

8th lecture on November 21, 2019. The lecture was cancelled because of the
Open Day at MFF UK.
However, here are lecture notes for the 8th lecture .

9th lecture on November 28, 2019. Series of functions, power series.
Lecture notes for the 9th lecture .

10th lecture on December 5, 2019. Two results on power series, Fourier series
(no proofs). Lecture notes for the 10th lecture .

11th lecture on December 12, 2019. Chapter 3: Introduction to
complex analysis. Holomorphic and analytic function, four differences, integration.
Lecture notes for the 11th lecture (updated on December 28).

12th lecture on December 19, 2019.
The constant rho=2pi.i, the Cauchy-Goursat theorem, the functional integral. Lecture notes for the
12th lecture .

13th lecture on January 9, 2020.
The Cauchy formulas, proofs of Theorems 1, 2 and 3, summing n^{-2k}. Lecture notes for the
13th lecture (updated at Jan 9).

January 2020