MY-ASSIGNMENTEXPERT™可以为您提供math.wustl Math5052 Functional Analysis信息论课程的代写代考和辅导服务！
这是圣路易斯华盛顿大学泛函分析课程的代写成功案例。
Math5052课程简介
Topics. This will be the second semester of a two semester graduate-level introduction to the theory of measure and integration in abstract and Euclidean spaces. Math 5051 and 5052 form the basis for the Ph.D. qualifying exam in analysis.
Prerequisites. Math 5051, or permission of instructor.
Time. Classes meet Mondays, Wednesdays, and Fridays, 10:00 am to 11:00 am, in Cupples I Hall, room 218.
Text. The lectures will follow the book Real Analysis for Graduate Students, Version 2.1, by Richard F. Bass. ISBN-13: 978-1502514455
This textbook was also used in Math 5051.
Note that, although a PDF version is freely available, the printed version is cheap and handy to have at times when computers are not available.
Prerequisites
Tests. There will be one midterm examination on Wednesday, March 9th, in class.
There will be a cumulative final examination, emphasizing later material, on Friday, May 6th, 2016 at 10:00am-12:00pm in Room 199.
Students may choose to take the real analysis qualifying examination at that date instead, which will last from 10:00am until 1:00pm in the same location.
No electronic devices will be allowed during these tests.
Grading. One grade will be assigned for all homework, one for the midterm, and one for the final examination. These grades will contribute as follows to the course grade: Homework 50%, Midterm 20%, Final 30%. Students taking the Cr/NCr or P/F options will need a grade of D or better to pass.
Math5052 Functional Analysis HELP（EXAM HELP， ONLINE TUTOR）
Let $A$ be the set of real-valued continuous functions on $[0,1]$ such that
$$
\int_0^{1 / 2} f(x) d x-\int_{1 / 2}^1 f(x) d x=1 .
$$
Prove that $A$ is a closed convex subset of $C([0,1])$, but there does not exist $f \in A$ such that $|f|=$ $\inf _{g \in A}|g|$.
Let $A_n$ be the subset of the real-valued continuous functions on $[0,1]$ given by
$$
A_n \stackrel{\text { def }}{=}{f:(\exists x \in[0,1])(\forall y \in[0,1])|f(x)-f(y)| \leq n|x-y|} .
$$
a. Prove that $A_n$ is nowhere dense in $C([0,1])$.
b. Prove that there exist functions $f \in C([0,1])$ which are nowhere differentiable on $[0,1]$, namely $f^{\prime}(x)$ does not exist at any point $x \in[0,1]$.
Let $X$ be a linear space and let $E \subset X$ be a convex set with $0 \in E$. Define a non-negative function $\rho: X \rightarrow \mathbf{R}$ by
$$
\rho(x) \stackrel{\text { def }}{=} \inf \left{t>0: t^{-1} x \in E\right}
$$
with the convention that $\rho(x)=\infty=\inf \emptyset$ if no $t>0$ gives $t^{-1} x \in E$. This called the Minkowski functional defined by $E$.
a. Show that $\rho$ is a sublinear functional, namely it satisfies $\rho(0)=0, \rho(x+y) \leq \rho(x)+\rho(y)$, and $\rho(\lambda x)=\lambda \rho(x)$ for all $x, y \in X$ and all $\lambda>0$.
b. Suppose in addition that $X$ is a normed linear space and $E$ is an open convex set containing 0 . Prove that the Minkowski functional defined by $E$ is finite at every $x \in X$ and that $x \in E$ if and only if $\rho(x)<1$.
Let $X$ be a normed linear space, let $E \subset X$ be an open convex set containing 0 , and let $\rho: X \rightarrow \mathbf{R}$ be the Minkowski functional defined by $E$. (See exercise 18 part a.) Prove that $\rho$ is continuous on $X$.
Let $X$ be a linear space and let $\rho: X \rightarrow \mathbf{R}$ be a sublinear functional. Suppose that $M$ is a subspace of $X$ and $f: M \rightarrow \mathbf{R}$ is a linear functional dominated by $\rho$, namely
$$
f(x) \leq \rho(x), \quad x \in M
$$
Prove that there exists a linear functional $F: X \rightarrow \mathbf{R}$ that satisfies $F(x)=f(x)$ for all $x \in M$ and $F(x) \leq \rho(x)$ for all $x \in X$.
NOTE: this implies the Hahn-Banach theorem, 18.5 on textbook p.173, in the special case $\rho(x) \stackrel{\text { def }}{=}|x|$.
Let $X$ be a Banach space, let $A \subset X$ be an open convex set, and let $B \subset X$ be a convex set disjoint from $A$. Prove that there exists a bounded real-valued linear functional $f$ and a constant $s \in \mathbf{R}$ such that $f(a)<s \leq f(b)$ for all $a \in A$ and all $b \in B$.
Hint: Consider the difference set $E=A-B+\left(a_0-b_0\right)$ for fixed $a_0 \in A, b_0 \in B$, and apply exercises 18,20 , and 21 .
MY-ASSIGNMENTEXPERT™可以为您提供STANFORD EE276/STATS376A: INFORMATION THEORY信息论课程的代写代考和辅导服务！
