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# 数学代写|拓扑学代写TOPOLOGY代考|MAST31023 THE EXTENDED STONE-WEIERSTRASS THEOREMS

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## 数学代写|拓扑学代写TOPOLOGY代考|THE EXTENDED STONE-WEIERSTRASS THEOREMS

Let $X$ be a locally compact Hausdorff space. Our present purpose is to generalize the theorems of Sec. 36 to this context.

A real or complex function $f$ defined on $X$ is said to vanish at infinity if for each $\epsilon>0$ there exists a compact subspace $C$ of $X$ such that $|f(x)|<\epsilon$ for every $x$ outside of $C$. On the real line, for instance, the functions $f$ and $g$ defined by $f(x)=e^{-x^{2}}$ and $g(x)=\left(x^{2}+1\right)^{-1}$ have this property, but the non-zero constant functions do not. It is easy to see that if $X$ is compact, then every real or complex function defined on $X$ vanishes at infinity, so in this case the requirement that a function vanish at infinity is no restriction at all.

We denote by $\mathcal{C}{0}(X, R)$ the set of all continuous real functions defined on $X$ which vanish at infinity. $\mathcal{C}{0}(X, C)$ is defined similarly. If $f$ is a function in one of these sets, then since $|f(x)|<\epsilon$ outside of some compact subspace $C$ of $X$, and $f$ is bounded on $C, f$ is necessarily bounded on all of $X . \quad$ It follows from this that $\mathcal{C}{0}(X, R) \subseteq \mathcal{e}(X, R)$ and $\mathcal{C}{0}(X, C) \subseteq \mathcal{C}(X, C)$. Further, the remark in the preceding paragraph shows that when $X$ is compact we have equality in each case.

Lemma. $\mathcal{C}{0}(X, R)$ and $\mathcal{C}{0}(X, C)$ are closed subalgebras of $\mathcal{C}(X, R)$ and $\mathcal{C}(X, C)$

PRoof. We first show that $\mathcal{C}{0}(X, R)$ is a closed subset of $\mathcal{C}(X, R)$. It suffices to show that if $f$ is a function in $\mathcal{C}(X, R)$ which is in the closure of $\mathcal{C}{0}(X, R)$, then $f$ vanishes at infinity. Let $\epsilon>0$ be given. Since $f$ is in the closure of $\mathcal{C}{0}(X, R)$, there exists a function $g$ in $\mathcal{C}{0}(X, R)$ such that $|f-g|<\epsilon / 2$, and this implies that $|f(x)-g(x)|<\epsilon / 2$ for all $x$. The function $g$ vanishes at infinity, so there exists a compact subspace $C$ of $X$ such that $|g(x)|<\epsilon / 2$ for all $x$ outside of $C$. It now follows at once that $|f(x)|=|[f(x)-g(x)]+g(x)| \leq|f(x)-g(x)|+|g(x)|<\epsilon / 2+\epsilon / 2=\epsilon$ for all $x$ outside of $C$, so $f$ vanishes at infinity. The same argument shows that $\mathcal{C}_{0}(X, C)$ is a closed subset of $\mathcal{C}(X, C)$.

## 数学代写|拓扑学代写TOPOLOGY代考|GROUPS

We begin by considering two familiar algebraic systems, each of which is a group, with a view to pointing out those features common to both which are set forth abstractly in the general concept of a group.
We first observe that the set $R$ of all real numbers, together with the operation of ordinary addition, has the following properties: the sum of any two numbers in $R$ is a number in $R$ ( $R$ is closed under addition); if $x, y, z$ are any three numbers in $R$, then $x+(y+z)=(x+y)+z$ (addition is associative); there is present in $R$ a special number, namely 0 , with the property that $x+0=0+x=x$ for every $x$ in $R$ ( $R$ contains an additive identity element); and to each number $x$ in $R$ there corresponds another number in $R$, its negative $-x$, with the property that $x+(-x)=(-x)+x=0(R$ contains additive inverses).

It is equally clear that the set $P$ of all positive real numbers, together with the operation of ordinary multiplication, has the following corresponding properties: the product of any two numbers in $P$ is a number in $P(P$ is closed under multiplication); if $x, y, z$ are any three numbers in $P$, then $x(y z)=(x y) z$ (multiplication is associative); there is present in $P$ a special number, namely 1 , with the property that $x 1=1 x=x$ for every $x$ in $P$ ( $P$ contains a multiplicative identity element); and to each number $x$ in $P$ there corresponds another number in $P$, its reciprocal $1 / x=x^{-1}$, with the property that $x x^{-1}=x^{-1} x=1$ ( $P$ contains multiplicative inverses).

## 数学代写|拓扑学代写TOPOLOGY代 考|THE EXTENDED STONEWEIERSTRASS THEOREMS

$|f(x)|=|[f(x)-g(x)]+g(x)| \leq|f(x)-g(x)|+|g(x)|<\epsilon / 2+\epsilon / 2=\epsilon$ 对所有人 $x$ 在 外面 $C$ ，所以 f消失在无穷远。同样的论证表明 $\mathcal{C}_{0}(X, C)$ 是的闭子集 $\mathcal{C}(X, C)$.

## 数学代写|拓扑学代写TOPOLOGY代 考|GROUPS

$x+(y+z)=(x+y)+z$ (加法是关联的) ；存在于 $R$ 一个特殊的数字，即 0 ，其属性 为 $x+0=0+x=x$ 对于每个 $x$ 在 $R(R$ 包含一个附加的标识元溸) ; 和每个昊码 $x$ 在 $R$ 有 对应的另一个数字 $R$, 其负 $-x$, 具有以下性质 $x+(-x)=(-x)+x=0(R$ 包含加法 逆)。

$x 1=1 x=x$ 对于每个 $x$ 在 $P(P$ 包含一个乘法恒等元）；和每个号码 $x$ 在 $P$ 有对应的另一 个数字 $P$, 它的倒数 $1 / x=x^{-1}$, 具有以下性质 $x x^{-1}=x^{-1} x=1(P$ 包含乘法逆元 $)$ 。

## 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。