## 数学代写|数学分析代写Mathematical Analysis代考|MATH2023

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical 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。我们的专家在代写数学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis相关的作业也就用不着说。

## 数学代写|数学分析代写Mathematical Analysis代考|Infinitesimal and infinite functions

Definition 5.8 Let $f$ be a function defined in a neighbourhood of c, except possibly at $c$. Then $f$ is said infinitesimal (or an infinitesimal) at $c$ if
$$\lim {x \rightarrow c} f(x)=0,$$ i.e., if $f=o(1)$ for $x \rightarrow c$. The function $f$ is said infinite at $c$ if $$\lim {x \rightarrow c} f(x)=\infty$$
Let us introduce the following terminology to compare two infinitesimal or infinite maps.
Definition 5.9 Let $f, g$ be two infinitesimals at $c$.
If $f \asymp g$ for $x \rightarrow c, f$ and $g$ are said infinitesimals of the same order.
If $f=o(g)$ for $x \rightarrow c, f$ is called infinitesimal of bigger order than $g$.
If $g=o(f)$ for $x \rightarrow c, f$ is called infinitesimal of smaller order than $g$.
If none of the above are satisfied, $f$ and $g$ are said non-comparable infinitesimals.

Definition 5.10 Let $f$ and $g$ be two infinite maps at $c$. If $f \asymp g$ for $x \rightarrow c, f$ and $g$ are said to be infinite of the same order. If $f=o(g)$ for $x \rightarrow c, f$ is called infinite of smaller order than $g$. If $g=o(f)$ for $x \rightarrow c, f$ is called infinite of bigger order than $g$. If none of the above are satisfied, the infinite functions $f$ and $g$ are said non-comparable.

## 数学代写|数学分析代写Mathematical Analysis代考|Asymptotes

We now consider a function $f$ defined in a neighbourhood of $+\infty$ and wish to study its behaviour for $x \rightarrow+\infty$. A remarkable case is that in which $f$ behaves as a polynomial of first degree. Geometrically speaking, this corresponds to the fact that the graph of $f$ will more and more look like a straight line. Precisely, we suppose there exist two real numbers $m$ and $q$ such that
$$\lim {x \rightarrow+\infty}(f(x)-(m x+q))=0,$$ or, using the symbols of Landau, $$f(x)=m x+q+o(1), \quad x \rightarrow+\infty .$$ We then say that the line $g(x)=m x+q$ is a right asymptote of the function $f$. The asymptote is called oblique if $m \neq 0$, horizontal if $m=0$. In geometrical terms condition (5.13) tells that the vertical distance $d(x)=|f(x)-g(x)|$ between the graph of $f$ and the asymptote tends to 0 as $x \rightarrow+\infty$ (Fig. 5.1). The asymptote’s coefficients can be recovered using limits: $$m=\lim {x \rightarrow+\infty} \frac{f(x)}{x} \quad \text { and } \quad q=\lim {x \rightarrow+\infty}(f(x)-m x) .$$ The first relation comes from (5.13) noting that $$0=\lim {x \rightarrow+\infty} \frac{f(x)-m x-q}{x}=\lim {x \rightarrow+\infty} \frac{f(x)}{x}-\lim {x \rightarrow+\infty} \frac{m x}{x}-\lim {x \rightarrow+\infty} \frac{q}{x}=\lim {x \rightarrow+\infty} \frac{f(x)}{x}-m,$$
while the second one follows directly from (5.13). The conditions (5.14) furnish the means to find the possible asymptote of a function $f$. If in fact both limits exist and are finite, $f$ admits $y=m x+q$ as a right asymptote. If only one of (5.14) is not finite instead, then $f$ will not have an asymptote.

# 数学分析代考

## 数学代写|数学分析代写Mathematical Analysis代考|Infinitesimal and infinite functions

$$\lim {x \rightarrow c} f(x)=0,$$即，如果$f=o(1)$为$x \rightarrow c$。函数$f$在$c$ if $$\lim {x \rightarrow c} f(x)=\infty$$处是无限的

## 数学代写|数学分析代写Mathematical Analysis代考|Asymptotes

$$\lim {x \rightarrow+\infty}(f(x)-(m x+q))=0,$$或者，使用朗道符号，$$f(x)=m x+q+o(1), \quad x \rightarrow+\infty .$$我们然后说，直线$g(x)=m x+q$是函数$f$的右渐近线。渐近线称为斜线$m \neq 0$，横线称为$m=0$。用几何术语来说，条件(5.13)告诉我们，$f$图形与渐近线之间的垂直距离$d(x)=|f(x)-g(x)|$趋近于0,$x \rightarrow+\infty$(图5.1)。渐近线的系数可以使用极限恢复:$$m=\lim {x \rightarrow+\infty} \frac{f(x)}{x} \quad \text { and } \quad q=\lim {x \rightarrow+\infty}(f(x)-m x) .$$第一个关系来自(5.13)，注意$$0=\lim {x \rightarrow+\infty} \frac{f(x)-m x-q}{x}=\lim {x \rightarrow+\infty} \frac{f(x)}{x}-\lim {x \rightarrow+\infty} \frac{m x}{x}-\lim {x \rightarrow+\infty} \frac{q}{x}=\lim {x \rightarrow+\infty} \frac{f(x)}{x}-m,$$

## 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。

## 数学代写|数学分析代写Mathematical Analysis代考|MATH2400

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical 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。我们的专家在代写数学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis相关的作业也就用不着说。

## 数学代写|数学分析代写Mathematical Analysis代考|Substitution theorem

The so-called Substitution theorem is important in itself for theoretical reasons, besides providing a very useful method to compute limits.
Theorem 4.15 Suppose a map $f$ admits limit
$$\lim {x \rightarrow c} f(x)=\ell,$$ finite or not. Let $g$ be defined on a neighbourhood of $\ell$ (excluding possibly the point $\ell$ ) and such that i) if $\ell \in \mathbb{R}, g$ is continuous at $\ell$; ii) if $\ell=+\infty$ or $\ell=-\infty$, the limit $\lim {y \rightarrow \ell} g(y)$ exists, finite or not.
Then the composition $g \circ f$ admits limit for $x \rightarrow c$ and
$$\lim {x \rightarrow c} g(f(x))=\lim {y \rightarrow \ell} g(y) .$$
Proof. Set $m=\lim _{y \rightarrow \ell} g(y)$ (noting that under $i$ ), $m=g(\ell)$ ). Given any neighbourhood $I(m)$ of $m$, by $i$ ) or ii) there will be a neighbourhood $I(\ell)$ of $\ell$ such that
$$\forall y \in \operatorname{dom} g, \quad y \in I(\ell) \quad \Rightarrow \quad g(y) \in I(m) .$$
Note that in case $i$ ) we can use $I(\ell)$ instead of $I(\ell) \backslash{\ell}$ because $g$ is continuous at $\ell$ (recall (3.7)), while $\ell$ does not belong to $I(\ell)$ for case $i i$ ). With such $I(\ell)$, assumption (4.9) implies the existence of a neighbourhood $I(c)$ of $c$ with
$$\forall x \in \operatorname{dom} f, \quad x \in I(c) \backslash{c} \quad \Rightarrow \quad f(x) \in I(\ell) .$$

Since $x \in \operatorname{dom} g \circ f$ means $x \in \operatorname{dom} f$ plus $y=f(x) \in \operatorname{dom} g$, the previous two implications now give
$$\forall x \in \operatorname{dom} g \circ f, \quad x \in I(c) \backslash{c} \quad \Rightarrow \quad g(f(x)) \in I(m) .$$
But $I(m)$ was arbitrary, so
$$\lim _{x \rightarrow e} g(f(x))=m .$$

## 数学代写|数学分析代写Mathematical Analysis代考|More fundamental limits. Indeterminate forms of exponential type

Consider the paramount limit (3.3). Instead of the sequence $a_n=\left(1+\frac{1}{n}\right)^n$, we look now at the function of real variable
$$h(x)=\left(1+\frac{1}{x}\right)^x .$$
It is defined when $1+\frac{1}{x}>0$, hence on $(-\infty,-1) \cup(0,+\infty)$. The following result states that $h$ and the sequence resemble each other closely when $x$ tends to infinity.
Property 4.20 The following limit holds
$$\lim {x \rightarrow \pm \infty}\left(1+\frac{1}{x}\right)^x=\mathrm{e}$$ Proof. $\leadsto$ The number e. By manipulating this formula we achieve a series of new fundamental limits. The substitution $y=\frac{x}{a}$, with $a \neq 0$, gives $$\lim {x \rightarrow \pm \infty}\left(1+\frac{a}{x}\right)^x=\lim {y \rightarrow \pm \infty}\left(1+\frac{1}{y}\right)^{a y}=\left[\lim {y \rightarrow \pm \infty}\left(1+\frac{1}{y}\right)^y\right]^a=\mathrm{e}^a .$$
In terms of the variable $y=\frac{1}{x}$ then,
$$\lim {x \rightarrow 0}(1+x)^{1 / x}=\lim {y \rightarrow \pm \infty}\left(1+\frac{1}{y}\right)^y=\mathrm{e} .$$
The continuity of the logarithm together with (4.11) furnish
$$\lim {x \rightarrow 0} \frac{\log _a(1+x)}{x}=\lim {x \rightarrow 0} \log a(1+x)^{1 / x}=\log _a \lim {x \rightarrow 0}(1+x)^{1 / x}=\log _a \mathrm{e}=\frac{1}{\log a}$$

# 数学分析代考

## 数学代写|数学分析代写Mathematical Analysis代考|Substitution theorem

$$\lim {x \rightarrow c} f(x)=\ell,$$有限与否。设$g$在$\ell$的邻域上定义(可能不包括$\ell$点)，并且满足i)如果$\ell \in \mathbb{R}, g$在$\ell$连续;Ii)如果$\ell=+\infty$或$\ell=-\infty$，则存在限制$\lim {y \rightarrow \ell} g(y)$，无论是否有限。

$$\lim {x \rightarrow c} g(f(x))=\lim {y \rightarrow \ell} g(y) .$$

$$\forall y \in \operatorname{dom} g, \quad y \in I(\ell) \quad \Rightarrow \quad g(y) \in I(m) .$$

$$\forall x \in \operatorname{dom} f, \quad x \in I(c) \backslash{c} \quad \Rightarrow \quad f(x) \in I(\ell) .$$

$$\forall x \in \operatorname{dom} g \circ f, \quad x \in I(c) \backslash{c} \quad \Rightarrow \quad g(f(x)) \in I(m) .$$

$$\lim _{x \rightarrow e} g(f(x))=m .$$

## 数学代写|数学分析代写Mathematical Analysis代考|More fundamental limits. Indeterminate forms of exponential type

$$h(x)=\left(1+\frac{1}{x}\right)^x .$$

$$\lim {x \rightarrow \pm \infty}\left(1+\frac{1}{x}\right)^x=\mathrm{e}$$证明。$\leadsto$数字e。通过操纵这个公式，我们获得了一系列新的基本极限。用$a \neq 0$替换$y=\frac{x}{a}$，得到$$\lim {x \rightarrow \pm \infty}\left(1+\frac{a}{x}\right)^x=\lim {y \rightarrow \pm \infty}\left(1+\frac{1}{y}\right)^{a y}=\left[\lim {y \rightarrow \pm \infty}\left(1+\frac{1}{y}\right)^y\right]^a=\mathrm{e}^a .$$

$$\lim {x \rightarrow 0}(1+x)^{1 / x}=\lim {y \rightarrow \pm \infty}\left(1+\frac{1}{y}\right)^y=\mathrm{e} .$$

$$\lim {x \rightarrow 0} \frac{\log _a(1+x)}{x}=\lim {x \rightarrow 0} \log a(1+x)^{1 / x}=\log _a \lim {x \rightarrow 0}(1+x)^{1 / x}=\log _a \mathrm{e}=\frac{1}{\log a}$$

## 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。

## 数学代写|数学分析代写Mathematical Analysis代考|MATH212

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical 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。我们的专家在代写数学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis相关的作业也就用不着说。

## 数学代写|数学分析代写Mathematical Analysis代考|Neighbourhoods

The process of defining limits and continuity leads to consider real numbers which are ‘close’ to a certain real number. In equivalent geometrical jargon, one considers points on the real line ‘in the proximity’ of a given point. Let us begin by making mathematical sense of the notion of neighbourhood of a point.

Definition 3.1 Let $x_0 \in \mathbb{R}$ be a point on the real line, and $r>0$ a real number. We call neighbourhood of $x_0$ of radius $r$ the open and bounded interval
$$I_r\left(x_0\right)=\left(x_0-r, x_0+r\right)=\left{x \in \mathbb{R}:\left|x-x_0\right|<r\right} .$$
Hence, the neighbourhood of 2 of radius $10^{-1}$, denoted $I_{10^{-1}}(2)$, is the set of real numbers lying between 1.9 and 2.1, these excluded. By understanding the quantity $\left|x-x_0\right|$ as the Euclidean distance between the points $x_0$ and $x$, we can then say that $I_r\left(x_0\right)$ consists of the points on the real line whose distance from $x_0$ is less than $r$. If we interpret $\left|x-x_0\right|$ as the tolerance in the approximation of $x_0$ by $x$, then $I_r\left(x_0\right)$ becomes the set of real numbers approximating $x_0$ with a better margin of precision than $r$.

Varying $r$ in the set of positive real numbers, while mantaining $x_0$ in $\mathbb{R}$ fixed, we obtain a family of neighbourhoods of $x_0$. Each neighbourhood is a proper subset of any other in the family that has bigger radius, and in turn it contains all neighbourhoods of lesser radius.

Remark 3.2 The notion of neighbourhood of a point $x_0 \in \mathbb{R}$ is nothing but a particular case of the analogue for a point in the Cartesian product $\mathbb{R}^d$ (hence the plane if $d=2$, space if $d=3$ ), presented in Definition 8.11.

The upcoming definitions of limit and continuity, based on the idea of neighbourhood, can be stated directly for functions on $\mathbb{R}^d$, by considering functions of one real variable as subcases for $d=1$. We prefer to follow a more gradual approach, so we shall examine first the one-dimensional case. Sect. 8.5 will be devoted to explaining how all this generalises to several dimensions.

## 数学代写|数学分析代写Mathematical Analysis代考|Limit of a sequence

Consider a real sequence $a: n \mapsto a_n$. We are interested in studying the behaviour of the values $a_n$ as $n$ increases, and we do so by looking first at a couple of examples.

Examples 3.4
i) Let $a_n=\frac{n}{n+1}$. The first terms of this sequence are presented in Table 3.1. We see that the values approach 1 as $n$ increases. More precisely, the real number 1 can be approximated as well as we like by $a_n$ for $n$ sufficiently large. This clause is to be understood in the following sense: however small we fix $\varepsilon>0$, from a certain point $n_{\varepsilon}$ onwards all values $a_n$ approximate 1 with a margin smaller that $\varepsilon$.
The condition $\left|a_n-1\right|<\varepsilon$, in fact, is tantamount to $\frac{1}{n+1}<\varepsilon$, i.e., $n+1>\frac{1}{\varepsilon}$; thus defining $n_{\varepsilon}=\left[\frac{1}{\varepsilon}\right]$ and taking any natural number $n>n_{\varepsilon}$, we have $n+1>$ $\left[\frac{1}{\varepsilon}\right]+1>\frac{1}{\varepsilon}$, hence $\left|a_n-1\right|<\varepsilon$. In other words, for every $\varepsilon>0$, there exists an $n_{\varepsilon}$ such that
$$n>n_{\varepsilon} \quad \Rightarrow \quad\left|a_n-1\right|<\varepsilon .$$ Looking at the graph of the sequence (Fig.3.3), one can say that for all $n>n_{\varepsilon}$ the points $\left(n, a_n\right)$ of the graph lie between the horizontal lines $y=1-\varepsilon$ and $y=1+\varepsilon$.

# 数学分析代考

## 数学代写|数学分析代写Mathematical Analysis代考|Neighbourhoods

$$I_r\left(x_0\right)=\left(x_0-r, x_0+r\right)=\left{x \in \mathbb{R}:\left|x-x_0\right|<r\right} .$$

## 数学代写|数学分析代写Mathematical Analysis代考|Limit of a sequence

i)让$a_n=\frac{n}{n+1}$。表3.1给出了这个序列的第一项。我们看到，随着$n$的增加，这些值趋于1。更准确地说，当$n$足够大时，实数1可以用$a_n$近似。该条款应在以下意义上理解:无论我们修复$\varepsilon>0$多么小，从某一点$n_{\varepsilon}$开始，所有值$a_n$都近似于1，边际小于$\varepsilon$。

$I$中的每个网格将$I$的一个分区定义为一个封闭区间$\mathcal{P}=\left{\left[x_0, x_1\right],\left[x_1, x_2\right], \ldots,\left[x_{k-1}, x_k\right]\right}$的有限集合。我们不区分$I$中的网格和它生成的分区。我们还用定义$I$的序列$\left{x_0, \ldots, x_k\right}$来表示它的分区。我们说分区$\mathcal{P}^{\prime}=\left{y_0, \ldots, y_m\right}$是分区$\mathcal{P}=\left{x_0, \ldots, x_k\right}$ if $\left{x_0, \ldots, x_k\right} \subseteq\left{y_0, \ldots, y_m\right}$的细化。这仅仅意味着$\mathcal{P}^{\prime}$是通过在一些(或所有)连续点$x_i$和$x_{i+1}$之间插入额外的网格点从$\mathcal{P}$获得的。注意，如果$\mathcal{P}^{\prime}$是$\mathcal{P}$的细化，那么$\mathcal{P}$中的每个间隔都是$\mathcal{P}^{\prime}$中间隔的并集。如果$\mathcal{P}$和$\mathcal{P}^{\prime}$是$[a, b]$的分区，那么$\mathcal{P}$和$\mathcal{P}^{\prime}$有一个共同的细化，即由$\operatorname{grid}\left{x_0, \ldots, x_k\right} \cup\left{y_0, \ldots, y_m\right}$生成的分区。

$$\Delta=\left{\sigma=J_1 \times J_2 \times \ldots \times J_n: J_i \in \mathcal{P}\right}$$

$$\Delta^{\prime}=\left{\sigma^{\prime}=J_1^{\prime} \times \ldots \times J_n^{\prime}: J_i^{\prime} \in \mathcal{P}_i^{\prime}\right}$$

## 数学代写|数学分析代写Mathematical Analysis代考|Measure Spaces

(a)如果$E \in \mathfrak{M}$，则$E^{\prime} \in \mathfrak{M}$;和
(b)如果$E_1, E_2 \in \mathfrak{M}$，则$E_1 \cup E_2 \in \mathfrak{M}$。

## 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。

## 数学代写|数学分析代写Mathematical Analysis代考|The Spectrum of an Operator

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical 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。我们的专家在代写数学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis相关的作业也就用不着说。

## 数学代写|数学分析代写Mathematical Analysis代考|The Spectrum of an Operator

The spectrum of a square matrix $A$ is simply its set of eigenvalues, and the eigenvalues of $A$ are easy to characterize. They are exactly the complex numbers $\lambda$ for which the matrix $A-\lambda I$ is not invertible. We recall the simple fact that $A-\lambda I$ is not invertible if and only if the linear operator $T$ it generates on $\mathbb{K}^n$ is not oneto-one, and this is the case if and only if $T$ in not onto.

The definition of the spectrum of an operator $T$ on an infinite-dimensional space is exactly the same as it is for a matrix. The stark distinction here is that not every point in the spectrum of an operator on an infinite-dimensional space is an eigenvalue. This is because such an operator may be one-to-one but not onto or conversely. See example 1. Thus the spectrum consists of two main parts: the complex numbers $\lambda$ for which $T-\lambda I$ is not one-to-one (the eigenvalues) and those for which $T-\lambda I$ is one to one but not onto. The spectrum of an operator $T$ often carries valuable information about $T$, and, in some cases, the eigenvalues of an operator and the corresponding eigenvectors completely define the operator.
Definition. A Banach algebra is a Banach space $X$ that is also an algebra with a multiplicative identity $I$ such that the norm satisfies the following additional assumptions:
(a) $|I|=1$, and
(b) $|S T| \leq|S| T |$ for all $S$ and $T$ in $X$.

We know that the set $\mathcal{L}(X)$ of bounded linear operators on a Banach space $X$ is a Banach space. In fact, $\mathcal{L}(X)$ is a Banach algebra with the composition of operators as the multiplication operation. The composition of two operators $S$ and $T$ is usually denoted by $S T$ rather than SoT. Property (a) is obvious, and property (b) follows from the inequalities $|(S T)(x)|=|S(T(x))| \leq|S||T(x)| \leq|S||T||x|$.
For the convenience of the reader, we list below the properties that make $\mathcal{L}(X)$ a Banach algebra: for operators $T, S, U \in \mathcal{L}(X)$ and all $a, b \in \mathbb{K}$,
(a) $(S T) U=S(T U)$
(b) $(a b) T=a(b T)$,
(c) $(T+S) U=T U+S U$ and $U(T+S)=U T+U S$,
(d) $|I|=1$, and
(e) $|S T| \leq|S||T|$.

## 数学代写|数学分析代写Mathematical Analysis代考|Adjoint Operators and Quotient Spaces

In section 3.7, we defined the adjoint of an operator on a finite-dimensional inner product space, and, in chapter 7 , we will study adjoints of operators on a Hilbert space. The definition of the adjoints on Banach spaces $X$ is more complicated. In fact, the adjoint of a bounded operator on a Banach space $X$ is a bounded operator on the dual space $X^$. Among other results, we prove that an operator $T$ and its adjoint, $T^$ have the same norm, the same spectrum, and the same spectral radius. We also study annihilators and quotient spaces. Little subsequent material rests on this section, and it is possible to study the remainder of the book independently of this section.

Notation. The duality bracket: Let $X$ be a Banach space. For $x \in X$ and $\lambda \in X^*$, we write $\langle x, \lambda\rangle$ for $\lambda(x)$. This is a notational convenience that also facilitates certain computations. In addition, the notation equalizes the roles of $X$ and $X^$. We already saw that $X$ acts on $X^$ in much the same way $X^*$ acts on $X$. See, for example, the construction leading up to theorem 6.4.8. Observe that $|\langle x, \lambda\rangle| \leq|x||\lambda|$, reminiscent of the Cauchy-Schwarz inequality. We revert to the traditional notation $\lambda(x)$ when convenient.

Theorem 6.6.1. Let $X$ be a Banach space, let $x \in X, \lambda \in X^$, and let $T \in \mathcal{L}(X)$. Then (a) $|\lambda|=\sup {|\langle x, \lambda\rangle|: x \in X,|x| \leq 1}$ (b) $|x|=|\hat{x}|=\sup \left{|\langle x, \lambda\rangle|: \lambda \in X^,|\lambda| \leq 1\right}$, and
(c) $|T|=\sup \left{|\langle T x, \lambda\rangle|: x \in X, \lambda \in X^*,|x| \leq 1,|\lambda| \leq 1\right}$.
Proof. (a) and (b) are previously established facts in new notation. To prove (c),
\begin{aligned} |T| & =\sup {|T x|:|x| \leq 1}=\sup {|x| \leq 1} \sup {|\lambda| \leq 1}|\langle T x, \lambda\rangle| \ & =\sup {|\langle T x, \lambda\rangle|:|x| \leq 1,|\lambda| \leq 1} . \end{aligned}

# 数学分析代考

## 数学代写|数学分析代写Mathematical Analysis代考|The Spectrum of an Operator

(a) $|I|=1$，以及
(b) $X$中所有的$S$和$T$均为$|S T| \leq|S| T |$。

(a) $(S T) U=S(T U)$
(b) $(a b) T=a(b T)$;
(c) $(T+S) U=T U+S U$和$U(T+S)=U T+U S$;
(d) $|I|=1$
(e) $|S T| \leq|S||T|$。

## 数学代写|数学分析代写Mathematical Analysis代考|Adjoint Operators and Quotient Spaces

(c) $|T|=\sup \left{|\langle T x, \lambda\rangle|: x \in X, \lambda \in X^*,|x| \leq 1,|\lambda| \leq 1\right}$。

\begin{aligned} |T| & =\sup {|T x|:|x| \leq 1}=\sup {|x| \leq 1} \sup {|\lambda| \leq 1}|\langle T x, \lambda\rangle| \ & =\sup {|\langle T x, \lambda\rangle|:|x| \leq 1,|\lambda| \leq 1} . \end{aligned}

## 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。

## 数学代写|数学分析代写Mathematical Analysis代考|Locally Compact Spaces

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical 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。我们的专家在代写数学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis相关的作业也就用不着说。

## 数学代写|数学分析代写Mathematical Analysis代考|Locally Compact Spaces

Without a doubt, $\mathbb{R}^n$ is the most important example of a locally compact Hausdorff space. We studied locally compact metric spaces briefly in section 4.7. In this section, we will see that locally compact Hausdorff spaces are regular (theorem 5.9.3); hence they have good separation properties. They are also very nearly normal. Compare theorems 5.9.2 and 5.6.3. The next section is the natural continuation of this one, where we show that every locally compact Hausdorff spaces can be embedded into a compact Hausdorff space in a special kind of way. We will take another journey into locally compact spaces in section 5.11 , where we establish Urysohn’s theorem for locally compact Hausdorff spaces and introduce the space of continuous, compactly supported functions on such spaces.

This section is the transitional section to the remaining three sections in this chapter. It may be bypassed on the first reading of the book because locally compact metric spaces (section 4.7) are sufficient for most of the rest of the book. Locally compact Hausdorff spaces are needed only in sections 8.4 and 8.7 , where frequent reference is made to the results in this section and sections 5.10 and 5.11 , and where certain theorems are extended from $\mathbb{R}^n$ to locally compact Hausdorff spaces.
Definition. A topological space $X$ is locally compact if, for every $x \in X$, there exists an open set $V$ such that $x \in V$ and $\bar{V}$ is compact. Thus every point is in the interior of a compact set.

We established in section 4.7 that $\mathbb{R}^n$ is locally compact and that $l^{\infty}$ is not. See theorem 6.1.5 for a far-reaching result. Also in section 4.7 , we showed that $\mathbb{Q}$ is not locally compact.

## 数学代写|数学分析代写Mathematical Analysis代考|Compactification

In this section, we show that a locally compact Hausdorff space $(X, \mathcal{J})$ can be embedded in a compact Hausdorff space $\left(X_{\infty}, \mathcal{J}{\infty}\right)$ in the manner described in theorem 5.10.1. In that theorem, the definition of the topology $\mathcal{J}{\infty}$ requires some explanation.

The prototypical and most important example of a locally compact Hausdorff space is $\mathbb{R}^n$. We focus here on $\mathbb{R}^2$, because the stereographic projection of the punctured sphere $\mathcal{S}*^2$ onto $\mathbb{R}^2$ is easy to visualize and provides an excellent motivation for the the definition of $\mathcal{J}{\infty}$. The stereographic projection has been known to mapmakers since the late sixteenth century, and it is reasonable to surmise that Alexandroff was aware of that projection when he invented the topology $\mathcal{J}_{\infty}$

It is clear that a compactification of the plane (more literally, its homeomorphic image $\mathcal{S}^2$ ) is the compact sphere $\mathcal{S}^2$, which contains $\mathcal{S}^2$ and a single additional point $N$. Some reflection reveals that there are two types of open subsets of the compact sphere:
(a) The open subsets of $\mathcal{S}^2$ that do not contain $N$ : These are in one-to-one correspondence (through the stereographic projection) with the open subsets of the usual topology of $\mathbb{R}^2$.
(b) The open subsets $U$ of $\mathcal{S}^2$ that contain the point $N$ : The complement $K=\mathcal{S}^2-U$ of such an open set is closed in $\mathcal{S}^2$. Since $\mathcal{S}^2$ is compact, $K$ is compact. Thus the open sets $U$ of this type are exactly the complements of compact subsets of the punctured sphere, which are in one-to-one correspondence with the compact subsets of $\mathbb{R}^2$.

# 数学分析代考

## 数学代写|数学分析代写Mathematical Analysis代考|Compactification

(a)不包含$N$的$\mathcal{S}^2$的开放子集:它们与$\mathbb{R}^2$通常拓扑的开放子集(通过立体投影)是一一对应的。
(b)包含点$N$的$\mathcal{S}^2$的开子集$U$:这个开集的补$K=\mathcal{S}^2-U$在$\mathcal{S}^2$中闭合。因为$\mathcal{S}^2$是紧凑的，所以$K$也是紧凑的。因此这种类型的开集$U$正是穿孔球紧子集的补集，它们与$\mathbb{R}^2$的紧子集是一一对应的。

## 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。

## 数学代写|数学分析代写Mathematical Analysis代考|Bases and Subbases

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical 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。我们的专家在代写数学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis相关的作业也就用不着说。

## 数学代写|数学分析代写Mathematical Analysis代考|Bases and Subbases

Some topologies are quite difficult to define directly, and it is frequently the case that we want to define a topology on a set $X$ that includes a certain collection ङ of subsets of $X$. The existence of such a topology is obvious because $\mathcal{P}(X)$ is such a topology. However, $\mathcal{P}(X)$ is useless because it is too large. This immediately suggests the question of finding the smallest topology $\mathcal{J}$ on $X$ that contains $\mathfrak{5}$. Fortunately, such a unique smallest topology $\mathcal{T}$ exists.

The reader may wonder what situations would compel us to “want” the members of $\subsetneq$ to be open. The prime such situation is when we need a certain class of functions from $X$ to another topological space $Y$ to be continuous, which is the overarching idea behind the definition of product and weak topologies. See sections 5.4 and 6.7 .

The set $\subsetneq$ in the above discussion is called a subbase for $\mathcal{T}$, and a closely connected concept is that of a base for the topology $\mathcal{T}$, which is our first definition. Bases and subbases have a wide range of applications. In addition to providing the means to define useful topologies, bases and subbases give us easy ways to prove the continuity of functions and to characterize closures. See theorems 5.2.2 and 5.3.1.
Definition. An open base for a topology $\mathcal{J}$ on a set $X$ is a collection $\mathfrak{B}$ of open subsets of $X$ such that every nonempty open subset in $X$ is the union of members of $\mathfrak{B}$. If $\mathfrak{B}$ is an open base for $\mathcal{T}$, we say that $\mathfrak{B}$ generates $\mathcal{T}$.

See problem 2 at the end of this section for an equivalent, more explicit formulation of the definition of an open base.

Example 1. The collection $\mathfrak{B}={(r, s): r, s \in \mathbb{Q}, r<s}$ is an open base for the usual topology on $\mathbb{R}$. This is because every open subset of $\mathbb{R}$ is the union of open bounded intervals, and any such interval is the union of members of $\boldsymbol{B}:(a, b)=$ $\cup{(r, s): r \in \mathbb{Q}, s \in \mathbb{Q}, a<r<s<b}$. See section 4.5 for a more general version of this example.

## 数学代写|数学分析代写Mathematical Analysis代考|Continuity

In section 4.3 , we studied the definition of local continuity of functions on metric spaces. It is clear that the $\epsilon-\delta$ definition provides no clues to generalizing the definition to the topological case. However, theorem 4.3.1 provides a metric-free characterization of local continuity which, with very slight changes, produces the following definition.
Definition. Let $X$ and $Y$ be topological spaces. A function $f: X \rightarrow Y$ is said to be continuous at a point $x_0 \in X$ if, for every open subset $V$ of $Y$ containing $f\left(x_0\right)$, $f^{-1}(V)$ contains an open neighborhood of $x_0$.
We point out here an important distinction between metric and general topologies. Theorem 4.3.2 established the fact that, in the metric case, continuity is equivalent to sequential continuity. This is not the case for a general topological space. See problem 11 at the end of this section.
As in the metric case, we can define a function from a topological space $X$ to another space $Y$ to be continuous if it is continuous at each point of $X$. However, theorem 4.3.3 suggests a more convenient, and widely used, definition of global continuity.
Definition. Let $\left(X, \mathcal{T}_X\right)$ and $\left(Y, \mathcal{T}_Y\right)$ be topological spaces. A function $f: X \rightarrow Y$ is said to be continuous if the inverse image of every open subset of $Y$ is an open subset of $X$. Symbolically, $V \in \mathcal{T}_Y$ implies $f^{-1}(V) \in \mathcal{T}_X$.
Continuity depends entirely on the topologies on $X$ and $Y$. Let $X=\mathbb{R}, \mathcal{T}_1$ be the discrete topology on $X$, and let $\mathcal{T}_2$ be the usual topology on $\mathbb{R}$. The identity function $I_X:\left(X, \mathcal{T}_1\right) \rightarrow\left(X, \mathcal{T}_2\right)$ is continuous, but the very same function $I_X:$ $\left(X, \mathcal{T}_2\right) \rightarrow\left(X, \mathcal{T}_1\right)$ is not continuous because not every subset of $\mathbb{R}$ is open in the usual topology of $\mathbb{R}$.

# 数学分析代考

## 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。