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）
Find a measure space $(X, \mathcal{A}, \mu)$, a subspace $Y$ of $L^1(X, \mu)$, and a bounded linear functional $f$ on $Y$ with norm 1 such that $f$ has two distinct extensions to $L^1(X, \mu)$ and each of the extensions has norm equal to 1.
Show that if $1 \leq p<\infty$, then $L^p([0,1])$ is separable, namely that there is a countable dense subset.
Show that $L^{\infty}([0,1])$ is not separable, namely that any dense subset must be uncountable.
For $k \geq 1$ and functions $f:[0,1] \rightarrow \mathbf{R}$ that are $k$ times differentiable, define
$$
|f|_{C^k} \stackrel{\text { def }}{=}|f|_{\infty}+\left|f^{\prime}\right|_{\infty}+\cdots+\left|f^{(k)}\right|_{\infty}
$$
where $f^{(k)}$ is the $k$ th derivative of $f$. Let $C^k([0,1])$ be the collection of $k$ times continuously differentiable functions $f$ with $|f|_{C^k}<\infty$.
Is $C^k([0,1])$ complete with respect to the norm $|\cdot|_{C^k}$ ?
Fix $\alpha \in(0,1)$. For a continuous function $f:[0,1] \rightarrow \mathbf{R}$, define
$$
|f|_{C^\alpha} \stackrel{\text { def }}{=} \sup {x \in[0,1]}|f(x)|+\sup {x \neq y \in[0,1]} \frac{|f(x)-f(y)|}{|x-y|^\alpha} .
$$
Let $C^\alpha([0,1])$ be the set of continuous functions $f$ with $|f|_{C^\alpha}<\infty$.
Is $C^\alpha([0,1])$ complete with respect to the norm $|\cdot|_{C^\alpha}$ ?
For positive integers $n$, let
$$
A_n \stackrel{\text { def }}{=}\left{f \in L^1([0,1]): \int_0^1|f(x)|^2 d x \leq n\right} .
$$
Show that each $A_n$ is a closed subset of $L^1([0,1])$ with empty interior.
Suppose $X$ and $Y$ are Banach spaces and $\mathcal{L}$ is the collection of bounded linear maps from $X$ into $Y$, with the usual operator norm:
$$
|L| \stackrel{\text { def }}{=} \sup _{|x|_X \leq 1}|L x|_Y
$$
Define $(L+M) x \stackrel{\text { def }}{=} L x+M x$ and $(c L) x=c(L x)$ for $L, M \in \mathcal{L}, x \in X$, and scalar $c$.
Prove that $\mathcal{L}$ is a Banach space.
NOTE: see Remark 18.10 on textbook p.178.
Set $A$ in a normed linear space is called convex if
$$
\lambda x+(1-\lambda) y \in A
$$
whenever $x, y \in A$ and $\lambda \in[0,1]$.
a. Prove that if $A$ is convex, then the closure of $A$ is convex.
b. Prove that the open unit ball in a normed linear space is convex. (The open unit ball is the set of $x$ such that $|x|<1$.)
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