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）
The unit ball in a normed linear space $V$ is called strictly convex if $|\lambda f+(1-\lambda) g|<1$ whenever $|f|=|g|=1, f \neq g \in V$, and $\lambda \in(0,1)$.
Let $(X, \mathcal{A}, \mu)$ be a measure space.
a. Prove that, if $1<p<\infty$, then the unit ball in $L^p(X, \mu)$ is strictly convex.
b. Prove that if $X$ contains two or more points, then the unit balls in $L^1(X, \mu)$ and $L^{\infty}(X, \mu)$ are not strictly convex.
Let $X$ be a metric space containing two or more points. Prove that the unit ball in $\mathcal{C}(X)$ is not strictly convex.
Let $f_n$ be a sequence of continuous functions on $\mathbf{R}$ that converge at every point. Prove that for every compact subset $K \subset \mathbf{R}$ there exists a number $M$ such that $\sup _n\left|f_n\right|$ is bounded by $M$ on that interval.
Suppose $|\cdot|_1$ and $|\cdot|_2$ are two norms on a vector space $X$ such that $|x|_1 \leq|x|_2$ for all $x \in X$, and suppose $X$ is complete with respect to both norms. Prove that there exists a positive constant $c$ such that
$$
|x|_2 \leq c|x|_1
$$
for all $x \in X$.
Suppose $X$ and $Y$ are Banach spaces.
a. Let $X \times Y$ be the set of ordered pairs $(x, y), x \in X, y \in Y$, with componentwise addition and multiplication by scalars. Define
$$
|(x, y)|_{X \times Y} \stackrel{\text { def }}{=}|x|_X+|y|_Y
$$
Prove that $X \times Y$ is a Banach space.
b. Let $L: X \rightarrow Y$ be a linear map such that if $x_n \rightarrow x$ in $X$ and $L x_n \rightarrow y$ in $Y$, then $y=L x$. Such a map is called a closed map. Let $G$ be the graph of $L$, defined by
$$
G \stackrel{\text { def }}{=}{(x, y) \in X \times Y: y=L x}
$$
Prove that $G$ is a closed subset of $X \times Y$, hence is complete.
c. Prove that the function $(x, L x) \mapsto x$ is continuous, injective, linear, and surjective from $G$ onto $X$.
d. Prove the closed graph theorem: If $L$ is a closed linear map from one Banach space to another (and hence by part b has a closed graph), then $L$ is a continuous map.
Let $X$ be the space of continuously differentiable functions on $[0,1]$ with the supremum norm and let $Y=C([0,1])$. Define $D: X \rightarrow Y$ by $D f=f^{\prime}$. Show that $D$ is a closed map but not a bounded one.
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！
