MY-ASSIGNMENTEXPERT™可以为您提供 cuhk.edu.hk Math540 Real Analysis实分析的代写代考和辅导服务！
这是香港中文大学实分析课程的代写成功案例。
MATH5011课程简介
Real and Complex Analysis, 3rd ed. W. Rudin, McGrawHill, New York 1966.
Measure Theory and Fine Properties of Functions, L.C. Evans and R.F. Gariepy, CRC Press 1992.
Real Analysis: Measure Theory, Integration and Hilbert Spaces, E.M. Stein and R. Shakarchi, Princeton Lectures in Analysis, Princeton 2005.
Real and Abstract Analysis, E. Hewitt and K. Stromberg, Graduate Texts in Mathematics, Springer-Verlag, New York 1975.
Prerequisites
Announcement
Here is the course outline Download file
The mid-term exam of Math 5011 will be held at LSB, LT6 next Friday, Oct 21, 2022, 18:30 pm-21:30pm. There will be no lecture in the morning of next Friday (Oct 21).
Course Description
Abstract integration theory; outer measures and Caratheodory’s construction, Borel, Radon and Hausdorff measures; positive linear functionals and Riesz representation theorem; Lp-spaces and their functional properties; and signed measures, Radon-Nikodym theorem and the dual of the space of continuous functions.
MATH5011 Real Analysis HELP（EXAM HELP， ONLINE TUTOR）
(3 in Rudin) Prove Proposition 1.15: The axioms of multiplication imply the following statements for all $x, y$, and $z$ in a field $F$.
(a) If $x \neq 0$ and $x y=x z$ then $y=z$.
(b) If $x \neq 0$ and $x y=x$ then $y=1$.
(c) If $x \neq 0$ and $x y=1$ then $y=1 / x$.
(d) If $x \neq 0$ then $1 /(1 / x)=x$.
Note that (b) and (c) assert the uniqueness of the multiplicative identity and the multiplicative inverse respectively.
(5 in Rudin) Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of all numbers $-x$, where $x \in A$. Prove that
$$
\inf A=-\sup (-A)
$$
Where the $\inf A$ is defined to be the greatest lower bound of $A$.
(8 in Rudin) Prove that no order can be defined in the complex field that turns it into an ordered field. Hint: -1 is a square.
(10 in Rudin) Suppose $z=a+i b, w=u+i v$, and
$$
a=\left(\frac{|w|+u}{2}\right)^{1 / 2}, \quad b=\left(\frac{|w|-u}{2}\right)^{1 / 2}
$$
Prove that $z^2=w$ if $v \geq 0$ and that $(\bar{z})^2=w$ if $v \leq 0$. Conclude that every complex number (with one exception!) has two complex square roots.
(13 in Rudin, this is often called the Reverse Traingle Inequality) If $x, y$ are complex, prove that
$$
|| x|-| y|| \leq|x-y|
$$
MY-ASSIGNMENTEXPERT™可以为您提供UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN MATH2940 linear algebra线性代数课程的代写代考和辅导服务！ 请认准MY-ASSIGNMENTEXPERT™. MY-ASSIGNMENTEXPERT™为您的留学生涯保驾护航。
