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数学代写|数学分析作业代写Mathematical Analysis代考|MTH131LR Product Spaces

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数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|Product Spaces

The Euclidean plane $\mathbb{R}^{2}$, as the product of two copies of $\mathbb{R}$, is the simplest example of a product space. We saw in section $4.3$ that the Euclidean metric in the plane, although the most natural, is equivalent to several other metrics, including the $\infty$-metric, which, according to the definition below, is the product metric on $\mathbb{R}^{2}$. It is only natural to expect that the product of two open intervals should be an open subset of $\mathbb{R}^{2}$, and the definition we adopt for the product metric smoothly guarantees that. When we identify the complex field with $\mathbb{R}^{2}$, the convergence of a complex sequence $z_{n}=x_{n}+i y_{n}$ is equivalent to the convergence of its real and imaginary parts in $\mathbb{R}$, and one expects that product metrics in general should extend this property. Not only does the product metric preserve the componentwise convergence in the factor spaces, it is characterized by it. You will see that the product metric is the weakest metric that guarantees componentwise convergence in the factor spaces. Additionally, we will show that the product metric admits the continuity of the projections on the factor spaces and, once again, is characterized by it. We therefore think of the product metric as the most economical metric that generalizes the properties of Euclidean space in relation to its factor spaces.

Let $\left{\left(X_{i}, d_{i}\right)\right}_{i=1}^{n}$ be a finite set of metric spaces, and let $X=\prod_{i=1}^{n} X_{i}=$ $\left{\left(x_{1}, \ldots, x_{n}\right): x_{i} \in X_{i}\right}$ be the Cartesian product of the underlying sets $X_{i}$.
Definition. The product metric $D$ on $X$ is defined by
$$D(x, y)=\max {1 \leq i \leq n} d{i}\left(x_{i}, y_{i}\right) .$$
Here $x=\left(x_{1}, \ldots, x_{n}\right)$ and $\left(y_{1}, \ldots, y_{n}\right)$ are points in $X$. The verification that $D$ is a metric is straightforward.

数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|Separable Spaces

Although the rigorous definition of the real line was a giant leap in the development of mathematics, it would not be nearly as useful an invention had it not been for the fact that it contains the rational numbers as a dense subset. Indeed, all practical computations, including machine calculations, are done exclusively using rational numbers. The simplicity of rational numbers is enhanced by their countability. Thus $\mathbb{Q}$ is numerous enough, simple enough, but not too enormous to be a useful approximation of $\mathbb{R}$. It is a reasonable quest to study metric spaces that contain a countable dense subset (of simpler elements). Such spaces are, by definition, separable. You will see that many (but not all) metric spaces are separable. The classical example is the space $\mathcal{C}[0,1]$. It is well known that (see section 4.8) the set of polynomials with rational coefficients, which is countable, is dense in $\mathcal{C}[0,1]$. What can be a nicer approximation of a continuous function than a rational polynomial! Separability of a metric space turns out to be equivalent to the existence of a countable collection of open sets that generate all open sets, which is an added benefit and an important characterization of separability.
Definition. A subset $A$ of a metric space $X$ is dense in $X$ if $\bar{A}=X$. By theorem 4.2.5, $A$ is dense in $X$ if and only if every point in $X$ is the limit of a sequence in A. Equivalently, $A$ is dense in $X$ if and only if for every $x \in X$ and every $\epsilon>0$, there is an element $a \in A$ such that $d(x, a)<\epsilon$.

Example 1. Given a function $f \in \mathcal{C}[0,1]$ and a number $\epsilon>0$, there exists a continuous, piecewise linear function $g$ such that $|f-g|_{\infty}<\epsilon$.

We use the uniform continuity of $f$ (see example 8 on section 1.2). Let $\delta>0$ be such that $|f(x)-f(y)|<\epsilon$ whenever $|x-y|<\delta$. Choose a natural number $n$ such that $1 / n<\delta$, and, for $0 \leq j \leq n$, let $x_{j}=j / n$. Define the function $g$ to be the continuous, piecewise linear function such that $g\left(x_{j}\right)=f\left(x_{j}\right)$ for $0 \leq j \leq n$. By construction, $|f-g|_{\infty}<\epsilon$. Observe that this example says that the space of continuous, piecewise linear functions is dense in $\mathcal{C}[0,1]$.

Definition. A metric space is separable if it contains a countable dense subset.

数学代写|数学分析作业代写 MATHEMATICAL ANALYSIS代 考|PRODUCT SPACES

$D(x, y)=\max 1 \leq i \leq n d i\left(x_{i}, y_{i}\right)$

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。