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# 数学代写|实分析代写Real Analysis代考|Definition and Examples

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## 数学代写|实分析代写Real Analysis代考|Definition and Examples

Let $X$ be a nonempty set. A function $d$ from $X \times X$, the set of ordered pairs of members of $X$, to the real numbers is a metric, or distance function, if
(i) $d(x, y) \geq 0$ always, with equality if and only if $x=y$,
(ii) $d(x, y)=d(y, x)$ for all $x$ and $y$ in $X$,
(iii) $d(x, y) \leq d(x, z)+d(z, y)$ for all $x, y$, and $z$, the triangle inequality. In this case the pair $(X, d)$ is called a metric space.

The real line $\mathbb{R}^1$ with metric $d(x, y)=|x-y|$ is the motivating example. Properties (i) and (ii) are apparent, and property (iii) is readily verified one case at a time according as $z$ is less than both $x$ and $y, z$ is between $x$ and $y$, or $z$ is greater than both $x$ and $y$.

We come to further examples in a moment. Particularly in the case that $X$ is a space of functions, a space may turn out to be almost a metric space but not to satisfy the condition that $d(x, y)=0$ implies $x=y$. Accordingly we introduce a weakened version of (i) as
(i’) $d(x, y) \geq 0$ and $d(x, x)=0$ always,
and we say that a function $d$ from $X \times X$ to the real numbers is a pseudometric if (i’), (ii), and (iii) hold. In this case, $(X, d)$ is called a pseudometric space.
Let $(X, d)$ be a pseudometric space. If $r>0$, the open ball of radius $r$ and center $x$, denoted by $B(r ; x)$, is the set of points at distance less than $r$ from $x$, namely
$$B(r ; x)={y \in X \mid d(x, y)<r} .$$

## 数学代写|实分析代写Real Analysis代考|Open Sets and Closed Sets

In this section we generalize the Euclidean notions of open set, closed set, neighborhood, interior, limit point, and closure so that they make sense for all pseudometric spaces, and we prove elementary properties relating these metricspace notions. In working with metric spaces and pseudometric spaces, it is often helpful to draw pictures as if the space in question were $\mathbb{R}^2$, even computing distances that are right for $\mathbb{R}^2$. We shall do that in the case of the first lemma but not afterward in this section. Let $(X, d)$ be a pseudometric space.

Lemma 2.4. If $z$ is in the intersection of open balls $B(r ; x)$ and $B(s ; y)$, then there exists some $t>0$ such that the open ball $B(t ; z)$ is contained in that intersection. Consequently the intersection of two open balls is open.

REMARK. Figure 2.2 shows what $B(t ; z)$ looks like in the metric space $\mathbb{R}^2$.
PROOF. Take $t=\min {r-d(x, z), s-d(y, z)}$. If $w$ is in $B(t ; z)$, then the triangle inequality gives
$$d(x, w) \leq d(x, z)+d(z, w)<d(x, z)+t \leq d(x, z)+(r-d(x, z))=r,$$
and hence $w$ is in $B(r ; x)$. Similarly $w$ is in $B(s ; y)$.

Proposition 2.5. The open sets of $X$ have the properties that
(a) $X$ and the empty set $\varnothing$ are open,
(b) an arbitrary union of open sets is open,
(c) any finite intersection of open sets is open.
PROOF. We know from Lemma 2.1 that a set is open if and only if it is the union of open balls. Then (b) is immediate, and (a) follows, since $X$ is the union of all open balls and $\varnothing$ is an empty union. For (c), it is enough to prove that $U \cap V$ is open if $U$ and $V$ are open. Write $U=\bigcup_\alpha B_\alpha$ and $V=\bigcup_\beta B_\beta$ as unions of open balls. Then $U \cap V=\bigcup_{\alpha, \beta}\left(B_\alpha \cap B_\beta\right)$, and Lemma 2.4 shows that $U \cap V$ is exhibited as the union of open balls. Thus $U \cap V$ is open.

## 数学代写|实分析代写Real Analysis代考|Definition and Examples

(i) $d(x, y) \geq 0$总是，当且仅当$x=y$，
(ii) $X$中所有的$x$和$y$为$d(x, y)=d(y, x)$;
(iii) $d(x, y) \leq d(x, z)+d(z, y)$对于所有的$x, y$，和$z$，三角不等式。在这种情况下$(X, d)$对被称为度量空间。

(i’)$d(x, y) \geq 0$和$d(x, x)=0$总是，

$$B(r ; x)={y \in X \mid d(x, y)<r} .$$

## 数学代写|实分析代写Real Analysis代考|Open Sets and Closed Sets

$$d(x, w) \leq d(x, z)+d(z, w)<d(x, z)+t \leq d(x, z)+(r-d(x, z))=r,$$

(a) $X$和空集$\varnothing$打开;
(b)开集的任意并是开的;
(c)任意开集的有限交是开的。

## 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。