Posted on Categories:Real analysis, 实分析, 数学代写

# 数学代写|实分析代写Real Analysis代考|Baire Category Theorem

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|实分析代写Real Analysis代考|Baire Category Theorem

A number of deep results in analysis depend critically on the fact that some metric space is complete. Already we have seen that the metric space $C(S)$ of bounded continuous scalar-valued functions on a metric space is complete, and we shall see as not too hard a consequence in Chapter XII that there exists a continuous periodic function whose Fourier series diverges at a point. One of the features of the Lebesgue integral in Chapter $\mathrm{V}$ will be that the metric spaces of integrable functions and of square-integrable functions, with their natural metrics, are further examples of complete metric spaces. Thus these spaces too are available for applications that make use of completeness.

The main device through which completeness is transformed into a powerful hypothesis is the Baire Category Theorem below. A closed set in a metric space is nowhere dense if its interior is empty. Its complement is an open dense set, and conversely the complement of any open dense set is closed nowhere dense.
EXAMPLE. A nontrivial example of a closed nowhere dense set is a Cantor set ${ }^4$ in $\mathbb{R}$. This is a set constructed from a closed bounded interval of $\mathbb{R}$ by removing an open interval in the middle of length a fraction $r_1$ of the total length with $0<r_1<1$, removing from each of the 2 remaining closed subintervals an open interval in the middle of length a fraction $r_2$ of the total length of the subinterval, removing from each of the 4 remaining closed subintervals an open interval in the middle of length a fraction $r_3$ of the total length of the interval, and so on indefinitely. The Cantor set is obtained as the intersection of the approximating sets. It is closed, being the intersection of closed sets, and it is nowhere dense because it contains no interval of more than one point. For the standard Cantor set, the starting interval is $[0,1]$, and the fractions are given by $r_1=r_2=\cdots=\frac{1}{3}$ at every stage. In general, the “length” of the resulting $\operatorname{set}^5$ is the product of the length of the starting interval and $\prod_{n=1}^{\infty}\left(1-r_n\right)$.

## 数学代写|实分析代写Real Analysis代考|Properties of C(S) for Compact Metric S

If $(S, d)$ is a metric space, then we saw in Proposition 2.44 that the vector space $B(S)$ of bounded scalar-valued functions on $S$, in the uniform metric, is a complete metric space. We saw also in Corollary 2.45 that the vector subspace $C(S)$ of bounded continuous functions is a complete subspace. In this section we shall study the space $C(S)$ further under the assumption that $S$ is compact. In this case Propositions 2.38 and 2.34 tell us that every continuous scalar-valued function on $S$ is automatically bounded and hence is in $C(S)$.

The first result about $C(S)$ for $S$ compact is a generalization of Ascoli’s Theorem from its setting in Theorem 1.22 for real-valued functions on a bounded interval $[a, b]$. The generalized theorem provides an insight that is not so obvious from the special case that $S$ is a closed bounded interval of $\mathbb{R}$. The insight is a characterization of the compact subsets of $C(S)$ when $S$ is compact, and it is stated precisely in Corollary 2.57 below. The relevant definitions for Ascoli’s Theorem are generalized in the expected way. Let $\mathcal{F}=\left{f_\alpha \mid \alpha \in A\right}$ be a set of scalar-valued functions on the compact metric space $S$. We say that $\mathcal{F}$ is equicontinuous at $x \in S$ if for each $\epsilon>0$, there is some $\delta>0$ such that $d(t, x)<\delta$ implies $|f(t)-f(x)|<\epsilon$ for all $f \in \mathcal{F}$. The set $\mathcal{F}$ of functions is pointwise bounded if for each $t \in[a, b]$, there exists a number $M_t$ such that $|f(t)| \leq M_t$ for all $f \in \mathcal{F}$. The set is uniformly equicontinuous on $S$ if it is equicontinuous at each point $x \in S$ and if the $\delta$ can be taken independent of $x$. The set is uniformly bounded on $S$ if it is pointwise bounded at each $t \in S$ and the bound $M_t$ can be taken independent of $t$; this last definition is consistent with the definition of a uniformly bounded sequence of functions given in Section 4.
Theorem 2.56 (Ascoli’s Theorem). Let $(S, d)$ be a compact metric space. If $\left{f_n\right}$ is a sequence of scalar-valued functions on $S$ that is equicontinuous at each point of $S$ and pointwise bounded on $S$, then
(a) $\left{f_n\right}$ is uniformly equicontinuous and uniformly bounded on $S$,
(b) $\left{f_n\right}$ has a uniformly convergent subsequence.

## 数学代写|实分析代写Real Analysis代考|Properties of C(S) for Compact Metric S

(a) $\left{f_n\right}$在$S$上均匀等连续，均匀有界;
(b) $\left{f_n\right}$具有一致收敛的子序列。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。