### 数学代写|实分析作业代写Real analysis代考|MATH1001

statistics-lab™ 为您的留学生涯保驾护航 在代写实分析Real analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写实分析Real analysis代写方面经验极为丰富，各种代写实分析Real analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|实分析作业代写Real analysis代考|Compact Sets and Their Properties

A very important role in continuous mathematics is played by the concept of compactness.
1.7.1. Definition. A set in a Hausdorff space is called compact (or compactum) if in every cover of this set by open sets one can pick a finite subcover:
It is clear from the definition that a set in a Hausdorff space is compact precisely when it is compact as a separate space with the induced topology. The property to be Hausdorff is not always included in the definition and is required here just for convenience of some subsequent formulations.

This definition is not intuitively motivated and may seem at the first glance to be too technical as compared to the intuitively convincing property of compactness of subsets of the real line formulated as the possibility of finding a convergent subsequence in every sequence. However, already a century long experience shows that the given definition (not equivalent to the definition in terms of sequences in case of general topological spaces, but coinciding with it in metric spaces) turns out to be much more fruitful and leads to a substantially more fruitful theory. A cover of a set by a family of open sets is called an open cover.
1.7.2. Proposition. (i) Any closed subset of a compact set is compact.
(ii) Any compact set in a Hausdorff space is closed.
(iii) The image of a compact set under a continuous mapping with values in a
Hausdorff space is compact.
(iv) Any infinite subset of a compact set has a limit point.
(v) Every continuous mapping from a compact metric space to a metric space is uniformly continuous.

## 数学代写|实分析作业代写Real analysis代考|Compactness Criteria

In the standard coordinate space $\mathbb{R}^n$ compact sets are precisely closed bounded sets. In calculus this fact is usually deduced from the case $n=1$, which in turn is established with the aid of basic properties of real numbers. In most of spaces interesting for applications the class of compact sets is strictly contained in the class of closed bounded sets. Hence it is important to have compactness criteria in concrete spaces. Here we consider three typical examples.
1.8.1. Theorem. $A$ set $K$ in the space $l^2$ is compact precisely when it is closed and bounded and satisfies the following condition:
$$\lim {N \rightarrow \infty} \sup {x \in K} \sum_{n=N}^{\infty} x_n^2=0 .$$
Proof. If $K$ is compact, then it is closed and bounded and for every $\varepsilon>0$ has a finite $\varepsilon$-net $a^1, \ldots, a^m$, where $a^i=\left(a_1^i, a_2^i, \ldots\right)$. Let us take $N$ such that $\sum_{n=N}^{\infty}\left|a_n^i\right|^2<\varepsilon^2$ for all $i \leqslant m$. We obtain $\sum_{n=N}^{\infty} x_n^2<4 \varepsilon^2$ for every $x \in K$, since there exists $i \leqslant m$ with $\sum_{n=1}^{\infty}\left|x_n-a_n^i\right|^2<\varepsilon^2$ and $x_n^2 \leqslant 2\left|x_n-a_n^i\right|^2+2\left|a_n^i\right|^2$. Conversely, if the indicated condition is fulfilled, then $K$ possesses a finite $\varepsilon$-net for every $\varepsilon>0$. Indeed, let $N$ be such that $\sup {x \in K} \sum{n=N+1}^{\infty} x_n^2<\varepsilon^2 / 4$. The set $K_N$ of points of the form $\pi_N x:=\left(x_1, \ldots, x_N, 0,0, \ldots\right)$, where $x \in K$, is an $\varepsilon / 2$-net for $K$ (since the distance between $x$ and $\pi_N x$ is not larger than $\varepsilon / 2$ ).

The set $K_N$ has a finite $\varepsilon / 2$-net (which will be a finite $\varepsilon$-net for $K$ ), since the projection of $K_N$ onto $\mathbb{R}^N$ is bounded by the boundedness of $K$ and hence has a finite $\varepsilon / 2$-net, which becomes an $\varepsilon / 2$-net for $K_N$ after adding zero coordinates starting from the $(N+1)$ th position.
1.8.2. Example. The set $E=\left{x \in l^2: \sum_{n=1}^{\infty} \alpha_n x_n^2 \leqslant 1\right}$, where $\alpha_n>0$ and $\alpha_n \rightarrow+\infty$, is compact in $l^2$. Indeed, it is easy to verify that it is closed and bounded. In addition,
$$\sup {x \in E} \sum{n=N}^{\infty} x_n^2 \leqslant \sup {x \in E} \sup {n \geqslant N} \alpha_n^{-1} \sum_{n=N}^{\infty} \alpha_n x_n^2 \leqslant \sup _{n \geqslant N} \alpha_n^{-1} \rightarrow 0$$
as $N \rightarrow \infty$. Hence the theorem proved above applies.

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|Compact Sets and Their Properties

1.7.1. 定义。豪斯多夫空间中的集合称为紧集（或紧集），如果在这个集合的每个开集覆盖中都可以选择一个有限子覆盖：

1.7.2. 主张。(i) 紧集的任何闭子集都是紧集的。
(ii) 豪斯多夫空间中的任何紧集都是闭集。(iii) 在Hausdorff 空间中具有值的连续映射下紧集的图像是紧的。
(iv) 紧集的任何无限子集都有一个极限点。
(v) 每个从紧度量空间到度量空间的连续映射是一致连续的。

## 数学代写|实分析作业代写Real analysis代考|Compactness Criteria

1.8.1. 定理。 $A$ 放 $K$ 在空间 $l^2$ 当它闭合且有界且满足以下条件时，它是紧致的:
$$\lim N \rightarrow \infty \sup x \in K \sum_{n=N}^{\infty} x_n^2=0$$

$$\sup x \in E \sum n=N^{\infty} x_n^2 \leqslant \sup x \in E \sup n \geqslant N \alpha_n^{-1} \sum_{n=N}^{\infty} \alpha_n x_n^2 \leqslant \sup _{n \geqslant N} \alpha_n^{-1} \rightarrow 0$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。