https://courses.pzhou.org/ 2026-04-17T11:32:35+00:00 https://courses.pzhou.org/ https://courses.pzhou.org/lib/tpl/dokuwiki/images/favicon.ico text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:hw:hw1&rev=1771684870&do=diff HW1 Due on gradescope on Jan 28 6pm. 0. Join our discord server, and setup your own homepage in the Students Area. 1. Try these exercises Tao-II, Ex 7.2.1 - 7.2.4, and Ex 7.4.1 - 7.4.4 Try not to look up the solutions. And write up the argument in your own way, and make your writing easy to follow. It is OK that you don't completely solve the problems, but do give each some thoughts. text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:hw:hw2&rev=1771684870&do=diff HW2 Due on gradescope next Friday 6pm. Please also submit on discord sometime around Wednesday. So far, we have climbed the 'one face' of the mountain Lebesgue measure, let's climb the other face. Here are some guides. Our goal is to start from the alternative definition of measurability (see text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:hw:hw3&rev=1771684870&do=diff HW 3 1. Read Pugh section 6.2, 6.4, 6.5. Read the proof of Theorem 21 and 26. 2. Pugh Ch 6, * Ex 3. Consider the special case of a diagonal line $\{y=x\}$ in $\R^2$. Can you prove directly that it has measure zero, i.e, using covering by boxes? You then may want to read Section 6.3 Theorem 9. Or, can you use today's zero slice theorem to prove it? text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:hw:hw4&rev=1771684870&do=diff HW4 In this week, we discussed Lebesgue integral, using the intuitive picture of undergraph. 0. write a short summary about Lebesgue integral, e.g how we define it, how does it compare with Riemann integrals. Share it on your homepage. 1. read Tao Analysis-II, 7.5 and 8.1. text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:hw:hw5&rev=1771684870&do=diff HW5 Tao-II: 8.2.7, 8.2.9, 8.2.10 Next week, we are going to consider Fubini's theorem in Tao 8.5 (see also Pugh section 6.7). You can read ahead. You may want to review the fact that, if a non-negative series is convergent, then any rearrangement of the series is convergent (the partial sum will form a monotone bounded sequence), hence a 'double series' of non-negative terms $\sum_n \sum_m a_{nm} = \sum_m \sum_n a_{nm} = \sum_{N=0}^\infty \sum_{n+m=N} a_{nm}$. text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:hw:hw6&rev=1771684870&do=diff HW 6 * Tao Ex 8.3.2, * Tao Ex 8.3.3 * Now we have covered the main results in Lebesgue measure theory, can you try to summarize the key steps and how to prove that? You can share it on your website. You can follow either Tao or Pugh's approach. text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:hw:hw7&rev=1771684870&do=diff HW 7 This week is about Vitali covering Lemma and its applications, Lebesgue density theorem. Pugh: Ch6: Ex 39, 48, 53, 58, 66 text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:hw:hw8&rev=1771684870&do=diff HW 8 1. Read appendix F about Littlewood's three principles, and write some comments about it in your webpage (for example, a summary of what this is about, or questions) 2. Do Pugh Ex 83 3. Let $(\R^n, | \cdot |_{1})$ be the normed vector space where $|(x_1, \cdots, x_n)|_{1}: = \sum_i |x_i| $. Let $T: \R^n \to \R^n$ be a linear operator, given by the matrix $T_{ij}$, that sends $(x_i)$ to $(y_j)$, where $y_i = \sum_j T_{ij} x_j$. How to compute $\|T \|$? text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:hw:hw9&rev=1771684870&do=diff HW 9 * Rudin Ex 8.6 * Rudin Ex 8.7 (Rudin's $D_i f = \partial f / \partial x_i$) * Show that, for any closed subset $E \In \R^2$, there is a continuous function $f: \R^2 \to \R$, such that $f^{-1}(0) = E$. (bonus, can you make $f$ a smooth function?) text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:hw:hw10&rev=1771684870&do=diff HW 10 Rudin Ch 9, * 12 (a,b,c) * 13 * 19 Pugh Ch 5. Ex 14, 24 text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:hw:hw11&rev=1771684870&do=diff HW 11 This weeks material is mostly conceptual, although the statement and proof in Pugh are all based on concrete formula. * Consider the generalized angular forms $\Omega_{n-1}$ defined on $\R^n \RM 0$ * For $n=2$, we define $\Omega_1 = |x|^{-2} (x_1 dx_2 - x_2 dx_1)$ text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:hw:hw12&rev=1771684870&do=diff HW 12 * Following Thursday's class, can you verify explicitly that any closed differential 2-form in $\R^3$ is exact? * (optional) read about Brouer's fixed point theorem in Pugh, and try Pugh Exercise 71, hairy ball theorem. text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:hw:start&rev=1771684870&do=diff Homeworks