Weekly outline

  • Introduction

    We shall discuss areas of modern mathematics, to gain some knowledge of current methods and trends in mathematics. We will also include necessary background, both theoretical and historical, to appreciate modern development. 

  • Recommended reading (will be updated)

    There are slides available for the course which summarize som of the topics (see below). As we progress, new materials will be posted in this section.

    For the moment, here are the lecture notes for the earlier version of this course in 2020. We will follow it, but with possible digressions according to our preferences.

    If you think you need to review basic set theory concepts and arguments, see the lecture notes on set theory below.

  • Topics

    3.10 - 17.10. We discussed the general mathematical context. We proved in detail that square root of 2 is not rational. We defined real numbers R as the collection of Dedekind cuts on rationals and sketched the proof that with this definition, reals are complete (or "continuous") in the sense that every bounded subset of R has the supremum. We proved that the powerset of natural numbers is strictly bigger than the set of natural numbers (see the lecture notes on set theory above for a review set-theoretical concepts).

    24.10. We discussed the notation of a metric space (X,d), and sketched proofs that N, Z, and N^2 have the same size, and that R^2 has the same size as R. We defined the notion of a Cauchy sequence in a metric space (X,d).

    31.10. We have further discussed the notion of metric and we introduced the notion of a separable metric space: (X,d) is separable if there is a countable H subset X such that for every x in X and every epsilon > 0, there is some h in H with d(x,h)<epsilon. We gave an argument that if (X,d) is a separable metric space then its size cannot be greater than 2^omega. Sketch of proof: for every x in X choose a sequence h_x = <h_n ; n < omega> of points in H such that d(x,h_n) < 1/n. Then if x \neq y, there must be some 1/n such that d(x,y) > 1/n. It follows that h_x \neq h_y. Since there are only 2^omega of subsets of the countable set H, X cannot have a larger size than 2^omega.

    7.11. (The class was cancelled). In preparation for the class on 14.11., try to come up with an argument for the following claim:

    Tentative theorem: If (X,d) is an infinite complete seperable metric space, then its size is 2^omega.

    1] By the result from class 31.10, X cannot be larger than 2^omega. Our tentative theorem says that it cannot be smaller, either.

    2] Recall that X is complete if every Cauchy sequence of point in X converges.

    14.11. We have finished the proof that every complete separable metric space without isolated points has size 2^omega. See pages 35-40 in the updated slides above for the proof. Note that our tentative theorem from 7.11 was not correct - with isolated points, it is possible there is a countable complete metric space. To repair the theorem we replaced "infinite" by "without isolated points" which makes it correct.

    21.11. Cancelled.

    28.11. We introduced vector spaces and discussed their basic properties.

    5.12. We discussed connection between properties of metric spaces, and topological spaces generated by a metric. See the PDF copy of the virtual whiteboard. And also the updated slides (7.12.2022), pages 48-53.

    12.12. We discussed the notion of a inner product which lead to the concept of the Hilbert space. See the updated lacture notes (17.12.)