Posted on Categories:Functional Analysis, 数学代写, 泛函分析

## 数学代写|泛函分析代写Functional Analysis代考|MATH510

avatest泛函分析functional analysis代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest™， 最高质量的matlab作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此matlab作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

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## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Examples of Variational Formulations

We shall study now various variational formulations for a model diffusion-convection-reaction problem. We will use both versions of the Closed Range Theorem (for continuous and for closed operators) to demonstrate that different formulations are simultaneously well posed.

Diffusion-Convection-Reaction Problem. Given a domain $\Omega \subset \mathbb{R}^N, N \geq 1$, we wish to determine $u(x), x \in \bar{\Omega}$, that satisfies the boundary-value problem:
\left{\begin{aligned} -\left(a_{i j} u_{, j}\right){, i}+\left(b_i u\right){, i}+c u & =f & & \text { in } \Omega \ u & =0 & & \text { on } \Gamma_1 \ a_{i j} u_{, j} n_j-b_i n_i u & =0 & & \text { on } \Gamma_2 \end{aligned}\right.
Coefficients $a_{i j}(x)=a_{j i}(x), b_i(x), c(x)$ represent (anisotropic) diffusion, advection, and reaction, and $f$ stands for a source term. We are using the Einstein summation convention, the simplified, engineering notation for derivatives,
$$u_{, i} \stackrel{\prime}{=} \frac{\partial u}{\partial x_i}$$
and $n_i$ denote components of the unit outward vector on $\Gamma$. For instance, we can think of $u(x)$ as the temperature at point $x$ and $f(x)$ as representing a heat source (sink) at $x . \Gamma_1, \Gamma_2$ represent two disjoint parts of the boundary. For simplicity of the exposition, we will deal with homogeneous boundary conditions only.

Additional Facts about Sobolev Spaces. We will need some additional fundamental facts about two energy spaces. The first is the already discussed classical $H^1$ Sobolev space consisting of all $L^2$-functions whose distributional derivatives are also functions, and they are $L^2$-integrable as well,
$$H^1(\Omega):=\left{u \in L^2(\Omega): \frac{\partial u}{\partial x_i} \in L^2(\Omega), i=1, \ldots, N\right}$$
The space is equipped with the norm,
$$|u|_{H^1}^2:=|u|^2+\sum_{i=1}^N\left|\frac{\partial u}{\partial x_i}\right|^2$$
where $|\cdot|$ denotes the $L^2$-norm. The second term constitutes a seminorm on $H^1(\Omega)$ and will be denoted by
$$|u|{H^1}^2:=\sum{i=1}^N\left|\frac{\partial u}{\partial x_i}\right|^2$$
The second space, $H(\operatorname{div}, \Omega)$, consists of all vector-valued $L^2$-integrable functions whose distributional divergence is also a function, and it is $L^2$-integrable,
$$H(\operatorname{div}, \Omega):=\left{\sigma=\left(\sigma_i\right)_{i=1}^N \in\left(L^2(\Omega)\right)^N: \operatorname{div} \sigma \in L^2(\Omega)\right}$$

The space is equipped with the norm,
$$|\sigma|_{H(\text { div })}^2:=|\sigma|^2+|\operatorname{div} \sigma|^2$$
where the $L^2$-norm of vector-valued functions is computed componentwise,
$$|\sigma|^2:=\sum_{i=1}^N\left|\sigma_i\right|^2$$
For both energy spaces, there exist trace operators that generalize the classical boundary trace for scalarvalued functions and boundary normal trace for vector-valued functions,
$$\left.u \rightarrow u\right|{\Gamma}, \quad \sigma \rightarrow \sigma_n=\left.\sum{i=1}^N \sigma_i\right|_{\Gamma} n_i$$

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Examples of Variational Formulations

\left{\begin{aligned} -\left(a_{i j} u_{, j}\right){, i}+\left(b_i u\right){, i}+c u & =f & & \text { in } \Omega \ u & =0 & & \text { on } \Gamma_1 \ a_{i j} u_{, j} n_j-b_i n_i u & =0 & & \text { on } \Gamma_2 \end{aligned}\right.

$$u_{, i} \stackrel{\prime}{=} \frac{\partial u}{\partial x_i}$$
$n_i$表示$\Gamma$上单位向外向量的分量。例如，我们可以认为$u(x)$是$x$点的温度，$f(x)$代表热源(汇)，$x . \Gamma_1, \Gamma_2$代表边界的两个不相交的部分。为了说明的简单性，我们只处理齐次边界条件。

$$H^1(\Omega):=\left{u \in L^2(\Omega): \frac{\partial u}{\partial x_i} \in L^2(\Omega), i=1, \ldots, N\right}$$

$$|u|{H^1}^2:=|u|^2+\sum{i=1}^N\left|\frac{\partial u}{\partial x_i}\right|^2$$

$$|u|{H^1}^2:=\sum{i=1}^N\left|\frac{\partial u}{\partial x_i}\right|^2$$

$$H(\operatorname{div}, \Omega):=\left{\sigma=\left(\sigma_i\right)_{i=1}^N \in\left(L^2(\Omega)\right)^N: \operatorname{div} \sigma \in L^2(\Omega)\right}$$

$$|\sigma|{H(\text { div })}^2:=|\sigma|^2+|\operatorname{div} \sigma|^2$$ 其中向量值函数的$L^2$ -范数是按分量计算的， $$|\sigma|^2:=\sum{i=1}^N\left|\sigma_i\right|^2$$

$$\left.u \rightarrow u\right|{\Gamma}, \quad \sigma \rightarrow \sigma_n=\left.\sum{i=1}^N \sigma_i\right|_{\Gamma} n_i$$

## 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。

Posted on Categories:Functional Analysis, 数学代写, 泛函分析

## 数学代写|泛函分析代写Functional Analysis代考|MA4551

avatest泛函分析functional analysis代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest™， 最高质量的matlab作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此matlab作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

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avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Riesz Representation Theorem

The properties of the topological dual of a Hilbert space constitute one of the most important collection of ideas in Hilbert space theory and in the study of linear operators. We recall from our study of topological duals of Banach spaces in the previous chapter that the dual of a Hilbert space $V$ is the vector space $V^{\prime}$ consisting of all continuous linear functionals on $V$. If $f$ is a member of $V^{\prime}$ we write, as usual,
$$f(\boldsymbol{v})=\langle f, \boldsymbol{v}\rangle$$
where $\langle\cdot, \cdot\rangle$ denotes the duality pairing on $V^{\prime} \times V$. Recall that $V^{\prime}$ is a normed space equipped with the dual norm
$$|f|_{V^{\prime}}=\sup {\boldsymbol{v} \neq 0} \frac{\langle f, \boldsymbol{v}\rangle}{|\boldsymbol{v}|_V}$$ Now, in the case of Hilbert spaces, we have a ready-made device for constructing linear and continuous functionals on $V$ by means of the scalar product $(\cdot, \cdot)_V$. Indeed, if $\boldsymbol{u}$ is a fixed element of $V$, we may define a linear functional $f_u$ directly by $$f{\boldsymbol{u}}(\boldsymbol{v}) \stackrel{\text { def }}{=}(\boldsymbol{v}, \boldsymbol{u})=\overline{(\boldsymbol{u}, \boldsymbol{v})} \quad \forall \boldsymbol{v} \in V$$
This particular functional depends on the choice $\boldsymbol{u}$, and this suggests that we describe this correspondence by introducing an operator $R$ from $V$ into $V^{\prime}$ such that
$$R \boldsymbol{u}=f_{\boldsymbol{u}}$$
We have by the definition
$$\langle R \boldsymbol{u}, \boldsymbol{v}\rangle=(\boldsymbol{v}, \boldsymbol{u})=\overline{(\boldsymbol{u}, \boldsymbol{v})} \quad \forall \boldsymbol{u}, \boldsymbol{v} \in V$$
Now, it is not clear at this point whether or not there might be some functionals in $V^{\prime}$ that cannot be represented by inner products on $V$. In fact, all we have shown up to now is that
$$R(V) \subset V^{\prime}$$

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|The Adjoint of a Linear Operator

In Sections 5.16 and 5.18 we examined the properties of the transpose of linear and both continuous and closed operators defined on Banach spaces. In the case of Hilbert spaces those ideas can be further specialized leading to the idea of (topologically) adjoint operators (recall Section 2.15 for a discussion of the same notion in finite-dimensional spaces).
We set the stage for this discussion by reviewing some notations. Let
$U, V$ be (complex) Hilbert spaces with scalar products $(\cdot, \cdot)_U$ and $(\cdot, \cdot)_V$, respectively.
$U^{\prime}, V^{\prime}$ denote the topological duals of $U$ and $V$.
$\langle\cdot, \cdot\rangle_U$ and $\langle\cdot, \cdot\rangle_V$ denote the duality pairings on $U^{\prime} \times U$ and $V^{\prime} \times V$.
$R_U: U \rightarrow U^{\prime}, R_V: V \rightarrow V^{\prime}$ be the Riesz operators for $U$ and $V$, respectively, i.e.,
\begin{aligned} & \left\langle R_U \boldsymbol{u}, \boldsymbol{w}\right\rangle=(\boldsymbol{w}, \boldsymbol{u})_U \forall \boldsymbol{w} \in U \quad \text { and } \ & \left\langle R_V \boldsymbol{v}, \boldsymbol{w}\right\rangle=(\boldsymbol{w}, \boldsymbol{v})_V \forall \boldsymbol{w} \in V \end{aligned}

(Topological) Adjoint of a Continuous Operator. Let $A \in \mathcal{L}(U, V)$, i.e., let $A$ be a linear and continuous operator from $U$ into $V$. Recall that the topological transpose operator $A^{\prime} \in \mathcal{L}\left(V^{\prime}, U^{\prime}\right)$ was defined as
$$A^{\prime} v^{\prime}=v^{\prime} \circ A \quad \text { for } \quad v^{\prime} \in V^{\prime}$$
or, equivalently,
$$\left\langle A^{\prime} v^{\prime}, \boldsymbol{u}\right\rangle=\left\langle v^{\prime}, A \boldsymbol{u}\right\rangle \quad \forall \boldsymbol{u} \in U v^{\prime} \in V^{\prime}$$
The transpose $A^{\prime}$ of operator $A$ operates on the dual $V^{\prime}$ into the dual $U^{\prime}$. Existence of the Riesz operators establishing the correspondence between spaces $U, V$ and their duals $U^{\prime}, V^{\prime}$ prompts us to introduce the so-called (topological) adjoint operator $A^$ operating directly on the space $V$ into $U$ and defined as the composition $$A^ \stackrel{\text { def }}{=} R_U^{-1} \circ A^{\prime} \circ R_V$$

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Riesz Representation Theorem

$$f(\boldsymbol{v})=\langle f, \boldsymbol{v}\rangle$$

$$|f|{V^{\prime}}=\sup {\boldsymbol{v} \neq 0} \frac{\langle f, \boldsymbol{v}\rangle}{|\boldsymbol{v}|_V}$$现在，在希尔伯特空间的情况下，我们有一个现成的装置来构造在$V$上的线性和连续泛函通过标量积$(\cdot, \cdot)_V$。的确，如果$\boldsymbol{u}$是$V$的一个固定元素，我们可以直接用$$f{\boldsymbol{u}}(\boldsymbol{v}) \stackrel{\text { def }}{=}(\boldsymbol{v}, \boldsymbol{u})=\overline{(\boldsymbol{u}, \boldsymbol{v})} \quad \forall \boldsymbol{v} \in V$$定义一个线性泛函$f_u$ 这个特殊的函数依赖于选择$\boldsymbol{u}$，这表明我们通过从$V$到$V^{\prime}$引入一个运算符$R$来描述这种对应关系，这样 $$R \boldsymbol{u}=f{\boldsymbol{u}}$$

$$\langle R \boldsymbol{u}, \boldsymbol{v}\rangle=(\boldsymbol{v}, \boldsymbol{u})=\overline{(\boldsymbol{u}, \boldsymbol{v})} \quad \forall \boldsymbol{u}, \boldsymbol{v} \in V$$

$$R(V) \subset V^{\prime}$$

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|The Adjoint of a Linear Operator

$U, V$分别为具有标量积$(\cdot, \cdot)_U$和$(\cdot, \cdot)_V$的(复)希尔伯特空间。
$U^{\prime}, V^{\prime}$表示$U$和$V$的拓扑对偶。
$\langle\cdot, \cdot\rangle_U$和$\langle\cdot, \cdot\rangle_V$表示$U^{\prime} \times U$和$V^{\prime} \times V$上的二元配对。
$R_U: U \rightarrow U^{\prime}, R_V: V \rightarrow V^{\prime}$分别为$U$和$V$的Riesz算符，即
\begin{aligned} & \left\langle R_U \boldsymbol{u}, \boldsymbol{w}\right\rangle=(\boldsymbol{w}, \boldsymbol{u})_U \forall \boldsymbol{w} \in U \quad \text { and } \ & \left\langle R_V \boldsymbol{v}, \boldsymbol{w}\right\rangle=(\boldsymbol{w}, \boldsymbol{v})_V \forall \boldsymbol{w} \in V \end{aligned}

$$A^{\prime} v^{\prime}=v^{\prime} \circ A \quad \text { for } \quad v^{\prime} \in V^{\prime}$$

$$\left\langle A^{\prime} v^{\prime}, \boldsymbol{u}\right\rangle=\left\langle v^{\prime}, A \boldsymbol{u}\right\rangle \quad \forall \boldsymbol{u} \in U v^{\prime} \in V^{\prime}$$

## 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。

Posted on Categories:Functional Analysis, 数学代写, 泛函分析

## 数学代写|泛函分析代写Functional Analysis代考|MAT4450

avatest泛函分析functional analysis代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest™， 最高质量的matlab作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此matlab作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

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## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Solvability of Linear Equations in Banach Spaces, the Closed Range Theorem

In this section we shall examine a collection of ideas that are very important in the abstract theory of linear operator equations on Banach spaces. They concern the solvability of equations of the form
$$A \boldsymbol{u}=\boldsymbol{f}, \quad A: U \longrightarrow V$$
where $A$ is a linear and continuous operator from a normed space $U$ into a normed space $V$, and $f$ is an element of $V$. Obviously, this equation can represent systems of linear algebraic equations, partial differential equations, integral equations, etc., so that general theorems concerned with its solvability are very important.
The question about the existence of solutions $\boldsymbol{u}$ to the equation above, for a given $\boldsymbol{f}$, can obviously be rephrased as
when does $\boldsymbol{f} \in \mathcal{R}(A)$ ?
where $\mathcal{R}(A)$ denotes the range of $A$. The characterization of the range $\mathcal{R}(A)$ is therefore crucial to our problem.
From the definition of the transpose
$$\left\langle\boldsymbol{v}^{\prime}, A \boldsymbol{u}\right\rangle=\left\langle A^{\prime} \boldsymbol{v}^{\prime}, \boldsymbol{u}\right\rangle \quad \forall \boldsymbol{u} \in U, \boldsymbol{v}^{\prime} \in V^{\prime}$$
we have that
$$\boldsymbol{v}^{\prime} \in \mathcal{N}\left(A^{\prime}\right) \quad \Leftrightarrow \quad \boldsymbol{v}^{\prime} \in \mathcal{R}(A)^{\perp}$$
which can be restated as
$$\mathcal{R}(A)^{\perp}=\mathcal{N}\left(A^{\prime}\right)$$
By the same reasoning
$$\mathcal{R}\left(A^{\prime}\right)^{\perp}=\mathcal{N}(A)$$
Combining these observations with Proposition 5.16.2, we arrive at the following conclusion.

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Quotient Normed Spaces

Quotient Normed Spaces. Let $U$ be a vector space and $M \subset U$ a subspace of $U$. In Chapter 2 we defined the quotient space $U / M$ consisting of equivalence classes of $\boldsymbol{u} \in U$ identified as affine subspaces of $U$ of the form
$$[\boldsymbol{u}]=\boldsymbol{u}+M={\boldsymbol{u}+\boldsymbol{v}: \boldsymbol{v} \in M}$$
If, in addition, $U$ is a normed space and $M$ is $c l o s e d$, the quotient space $U / M$ can be equipped with the norm
$$|[\boldsymbol{u}]|_{U / M} \stackrel{\text { def }}{=} \inf {\boldsymbol{v} \in[\boldsymbol{u}]}|\boldsymbol{v}|_U$$ Indeed, all properties of norms are satisfied: (i) $|[\boldsymbol{u}]|=0$ implies that there exists a sequence $\boldsymbol{v}_n \in[\boldsymbol{u}]$ such that $\boldsymbol{v}_n \rightarrow \mathbf{0}$. By closedness of $M$ and, therefore, of every equivalence class $[\boldsymbol{u}]$ (explain, why?), $\mathbf{0} \in[\boldsymbol{u}]$, which means that $[\boldsymbol{u}]=[\mathbf{0}]=M$ is the zero vector in the quotient space $U / M$. (ii) \begin{aligned} |\lambda[\boldsymbol{u}]| & =|[\lambda \boldsymbol{u}]| \ & =\inf {\lambda \boldsymbol{v} \in[\lambda \boldsymbol{u}]}|\lambda \boldsymbol{v}| \ & =|\lambda| \inf {\boldsymbol{v} \in[\boldsymbol{u}]}|\boldsymbol{v}|=|\lambda||[\boldsymbol{u}]| \end{aligned} (iii) Let $[\boldsymbol{u}],[\boldsymbol{v}] \in U / M$. Pick an arbitrary $\varepsilon>0$. Then, there exist $\boldsymbol{u}{\varepsilon} \in[\boldsymbol{u}]$ and $\boldsymbol{v}{\varepsilon} \in[\boldsymbol{v}]$ such that $$\left|\boldsymbol{u}{\varepsilon}\right| \leq|[\boldsymbol{u}]|_{U / M}+\frac{\varepsilon}{2} \text { and }\left|\boldsymbol{v}{\varepsilon}\right| \leq|[\boldsymbol{v}]|{U / M}+\frac{\varepsilon}{2}$$
Consequently
$$\left|\boldsymbol{u}{\varepsilon}+\boldsymbol{v}{\varepsilon}\right| \leq|[\boldsymbol{u}]|_{U / M}+|[\boldsymbol{v}]|_{U / M}+\varepsilon$$
But $\boldsymbol{u}{\varepsilon}+\boldsymbol{v}{\varepsilon} \in[\boldsymbol{u}+\boldsymbol{v}]$ and therefore taking the infimum on the left-hand side and passing to the limit with $\varepsilon \rightarrow 0$, we get the triangle inequality for the norm in $U / M$.

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Solvability of Linear Equations in Banach Spaces, the Closed Range Theorem

$$A \boldsymbol{u}=\boldsymbol{f}, \quad A: U \longrightarrow V$$

$\boldsymbol{f} \in \mathcal{R}(A)$什么时候?

$$\left\langle\boldsymbol{v}^{\prime}, A \boldsymbol{u}\right\rangle=\left\langle A^{\prime} \boldsymbol{v}^{\prime}, \boldsymbol{u}\right\rangle \quad \forall \boldsymbol{u} \in U, \boldsymbol{v}^{\prime} \in V^{\prime}$$

$$\boldsymbol{v}^{\prime} \in \mathcal{N}\left(A^{\prime}\right) \quad \Leftrightarrow \quad \boldsymbol{v}^{\prime} \in \mathcal{R}(A)^{\perp}$$

$$\mathcal{R}(A)^{\perp}=\mathcal{N}\left(A^{\prime}\right)$$

$$\mathcal{R}\left(A^{\prime}\right)^{\perp}=\mathcal{N}(A)$$

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Quotient Normed Spaces

$$[\boldsymbol{u}]=\boldsymbol{u}+M={\boldsymbol{u}+\boldsymbol{v}: \boldsymbol{v} \in M}$$

$$|[\boldsymbol{u}]|{U / M} \stackrel{\text { def }}{=} \inf {\boldsymbol{v} \in[\boldsymbol{u}]}|\boldsymbol{v}|_U$$的确，范数的所有性质都被满足:(i) $|[\boldsymbol{u}]|=0$暗示存在一个序列$\boldsymbol{v}_n \in[\boldsymbol{u}]$，使得$\boldsymbol{v}_n \rightarrow \mathbf{0}$。通过$M$的紧密性，因此，每个等价类$[\boldsymbol{u}]$(解释，为什么?)，$\mathbf{0} \in[\boldsymbol{u}]$，这意味着$[\boldsymbol{u}]=[\mathbf{0}]=M$是商空间$U / M$中的零向量。(ii) \begin{aligned} |\lambda[\boldsymbol{u}]| & =|[\lambda \boldsymbol{u}]| \ & =\inf {\lambda \boldsymbol{v} \in[\lambda \boldsymbol{u}]}|\lambda \boldsymbol{v}| \ & =|\lambda| \inf {\boldsymbol{v} \in[\boldsymbol{u}]}|\boldsymbol{v}|=|\lambda||[\boldsymbol{u}]| \end{aligned} (iii)让$[\boldsymbol{u}],[\boldsymbol{v}] \in U / M$。随便选一个$\varepsilon>0$。然后，存在$\boldsymbol{u}{\varepsilon} \in[\boldsymbol{u}]$和$\boldsymbol{v}{\varepsilon} \in[\boldsymbol{v}]$，使得$$\left|\boldsymbol{u}{\varepsilon}\right| \leq|[\boldsymbol{u}]|{U / M}+\frac{\varepsilon}{2} \text { and }\left|\boldsymbol{v}{\varepsilon}\right| \leq|[\boldsymbol{v}]|{U / M}+\frac{\varepsilon}{2}$$

$$\left|\boldsymbol{u}{\varepsilon}+\boldsymbol{v}{\varepsilon}\right| \leq|[\boldsymbol{u}]|{U / M}+|[\boldsymbol{v}]|{U / M}+\varepsilon$$

## 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。

Posted on Categories:Functional Analysis, 数学代写, 泛函分析

## 数学代写|泛函分析代写Functional Analysis代考|Closed Operators, Closed Graph Theorem

avatest泛函分析functional analysis代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest™， 最高质量的matlab作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此matlab作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

## avatest™帮您通过考试

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## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Closed Operators, Closed Graph Theorem

We begin with some simple observations concerning Cartesian products of normed spaces. First of all, recall that if $X$ and $Y$ are vector spaces, then the Cartesian product $X \times Y$ is also a vector space with operations defined by
\begin{aligned} \left(\boldsymbol{x}_1, \boldsymbol{y}_1\right)+\left(\boldsymbol{x}_2, \boldsymbol{y}_2\right) & \stackrel{\text { def }}{=}\left(\boldsymbol{x}_1+\boldsymbol{x}_2, \boldsymbol{y}_1+\boldsymbol{y}_2\right) \ \alpha(\boldsymbol{x}, \boldsymbol{y}) & \stackrel{\text { def }}{=}(\alpha \boldsymbol{x}, \alpha \boldsymbol{y}) \end{aligned}
where the vector additions and multiplications by a scalar on the right-hand side are those in the $X$ and $Y$ spaces, respectively.

If additionally $X$ and $Y$ are normed spaces with norms $|\cdot|_X$ and $|\cdot|_Y$, respectively, then $X \times Y$ may be equipped with a (not unique) norm of the form
$$|(\boldsymbol{x}, \boldsymbol{y})|= \begin{cases}\left(|\boldsymbol{x}|_X^p+|\boldsymbol{y}|_Y^p\right)^{\frac{1}{p}} & 1 \leq p<\infty \ \max \left{|\boldsymbol{x}|_X,|\boldsymbol{y}|_Y\right} & p=\infty\end{cases}$$
Finally, if $X$ and $Y$ are complete, then $X \times Y$ is also complete. Indeed, if $\left(\boldsymbol{x}_n, \boldsymbol{y}_n\right)$ is a Cauchy sequence in $X \times Y$, then $\boldsymbol{x}_n$ is a Cauchy sequence in $X$, and $\boldsymbol{y}_n$ is a Cauchy sequence in $Y$. Consequently both $\boldsymbol{x}_n$ and $\boldsymbol{y}_n$ have limits, say $\boldsymbol{x}$ and $\boldsymbol{y}$, and, therefore, by the definition of the norm in $X \times Y,\left(\boldsymbol{x}_n, \boldsymbol{y}_n\right) \rightarrow(\boldsymbol{x}, \boldsymbol{y})$. Thus, if $X$ and $Y$ are Banach spaces, then $X \times Y$ is a Banach space, too.

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Operators

Up to this point, all of the linear transformations from a vector space $X$ into a vector space $Y$ have been defined on the whole space $X$, i.e., their domain of definition coincided with the entire space $X$. In a more general situation, it may be useful to consider linear operators defined on a proper subspace of $X$ only (see Example 5.6.4). In fact, some authors reserve the name operator to such functions distinguishing them from transformations which are defined on the whole space.

Thus, in general, a linear operator $T$ from a vector space $X$ into a vector space $Y$ may be defined only on a proper subspace of $X$, denoted $D(T)$ and called the domain of definition of $T$, or concisely, the domain of $T:$
$$X \supset D(T) \ni \boldsymbol{x} \longrightarrow T \boldsymbol{x} \in Y$$
Note that in the case of linear operators, the domain $D(T)$ must be a vector subspace of $X$ (otherwise it would make no sense to speak of linearity of $T$ ).

Still, the choice of the domain is somehow arbitrary. Different domains with the same rule defining $T$ result formally in different operators in much the same fashion as functions are defined by specifying their domain, codomain, and the rule (see Chapter 1).

With every operator $T$ (not necessarily linear) we can associate its graph, denoted $G(T)$ and defined as graph $T=G(T) \stackrel{\text { def }}{=}{(\boldsymbol{x}, T \boldsymbol{x}): \boldsymbol{x} \in D(T)} \subset X \times Y$
(recall the discussion in Section 1.9).

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Closed Operators, Closed Graph Theorem

\begin{aligned} \left(\boldsymbol{x}_1, \boldsymbol{y}_1\right)+\left(\boldsymbol{x}_2, \boldsymbol{y}_2\right) & \stackrel{\text { def }}{=}\left(\boldsymbol{x}_1+\boldsymbol{x}_2, \boldsymbol{y}_1+\boldsymbol{y}_2\right) \ \alpha(\boldsymbol{x}, \boldsymbol{y}) & \stackrel{\text { def }}{=}(\alpha \boldsymbol{x}, \alpha \boldsymbol{y}) \end{aligned}

$$|(\boldsymbol{x}, \boldsymbol{y})|= \begin{cases}\left(|\boldsymbol{x}|_X^p+|\boldsymbol{y}|_Y^p\right)^{\frac{1}{p}} & 1 \leq p<\infty \ \max \left{|\boldsymbol{x}|_X,|\boldsymbol{y}|_Y\right} & p=\infty\end{cases}$$

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Operators

$$X \supset D(T) \ni \boldsymbol{x} \longrightarrow T \boldsymbol{x} \in Y$$

(回想1.9节的讨论)。

## 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。

Posted on Categories:Functional Analysis, 数学代写, 泛函分析

## 数学代写|泛函分析代写Functional Analysis代考|Space of Test Functions

avatest泛函分析functional analysis代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest™， 最高质量的matlab作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此matlab作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Space of Test Functions

Functions with Compact Support. Let $\Omega \subset \mathbb{R}^n$ be an open set and $u$ any real- (or complex-) valued function defined on $\Omega$. The closure of the set of all points $x \in \Omega$ for which $u$ takes non-zero values is called the support of $u$ :
$$\operatorname{supp} u \stackrel{\text { def }}{=} \overline{{x \in \Omega: u(x) \neq 0}}$$
Note that, due to the closure operation, the support of a function $u$ may include the points at which $u$ vanishes (see Fig. 5.1).

The collection of all infinitely differentiable functions defined on $\Omega$, whose supports are compact (i.e., bounded) and contained in $\Omega$, will be denoted as
$$C_0^{\infty}(\Omega) \stackrel{\text { def }}{=}\left{u \in C^{\infty}(\Omega): \operatorname{supp} u \subset \Omega, \quad \operatorname{supp} u \text { compact }\right}$$
Obviously, $C_0^{\infty}(\Omega)$ is a vector subspace of $C^{\infty}(\Omega)$.
Example 5.3.1
A standard example of a function in $C_0^{\infty}(\mathbb{R})$ is
$$\phi(x)= \begin{cases}\exp \left[1 /\left(x^2-a^2\right)\right] & |x|0) \ 0 & |x| \geq a\end{cases}$$

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|The Hahn-Banach Theorem

In this section we establish a fundamental result concerning the extension of linear functionals on infinitedimensional vector spaces, the famous Hahn-Banach theorem. The result will be obtained in a general setting of arbitrary vector spaces and later on specialized in a more specific context.

Sublinear Functionals. Let $V$ be a real vector space. A functional $p: V \rightarrow \mathbb{R}$ is said to be sublinear iff
(i) $p(\alpha \boldsymbol{u})=\alpha p(\boldsymbol{u}) \quad \forall \alpha>0$
(ii) $p(\boldsymbol{u}+\boldsymbol{v}) \leq p(\boldsymbol{u})+p(\boldsymbol{v}) \quad(p$ is subadditive $)$
for arbitrary vectors $\boldsymbol{u}$ and $\boldsymbol{v}$. Obviously, every linear functional is sublinear and every seminorm is sublinear as well.
THEOREM 5.4.1
(The Hahn-Banach Theorem)
Let $X$ be a real vector space, $p: X \rightarrow \mathbb{R}$ a sublinear functional on $X$, and $M \subset X$ a subspace of $X$. Consider $f: M \rightarrow \mathbb{R}$, a linear functional on $M\left(f \in M^*\right)$ dominated by $p$ on $M$, i.e.,
$$f(\boldsymbol{x}) \leq p(\boldsymbol{x}) \quad \forall \boldsymbol{x} \in M$$
Then, there exists a linear functional $F: X \rightarrow \mathbb{R}$ defined on the whole $X$ such that
(i) $\left.F\right|_M \equiv f$
(ii) $F(\boldsymbol{x}) \leq p(\boldsymbol{x}) \quad \forall \boldsymbol{x} \in X$
In other words, $F$ is an extension of $f$ dominated by $p$ on the whole $X$.

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Space of Test Functions

$$\operatorname{supp} u \stackrel{\text { def }}{=} \overline{{x \in \Omega: u(x) \neq 0}}$$

$$C_0^{\infty}(\Omega) \stackrel{\text { def }}{=}\left{u \in C^{\infty}(\Omega): \operatorname{supp} u \subset \Omega, \quad \operatorname{supp} u \text { compact }\right}$$

$C_0^{\infty}(\mathbb{R})$中函数的标准示例如下
$$\phi(x)= \begin{cases}\exp \left[1 /\left(x^2-a^2\right)\right] & |x|0) \ 0 & |x| \geq a\end{cases}$$

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|The Hahn-Banach Theorem

(i) $p(\alpha \boldsymbol{u})=\alpha p(\boldsymbol{u}) \quad \forall \alpha>0$
(ii) $p(\boldsymbol{u}+\boldsymbol{v}) \leq p(\boldsymbol{u})+p(\boldsymbol{v}) \quad(p$是次加性的$)$

(哈恩-巴拿赫定理)

$$f(\boldsymbol{x}) \leq p(\boldsymbol{x}) \quad \forall \boldsymbol{x} \in M$$

(i) $\left.F\right|_M \equiv f$
(ii) $F(\boldsymbol{x}) \leq p(\boldsymbol{x}) \quad \forall \boldsymbol{x} \in X$

## 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。

Posted on Categories:Functional Analysis, 数学代写, 泛函分析

## 数学代写|泛函分析代写Functional Analysis代考|Topological Properties of Metric Spaces

avatest泛函分析functional analysis代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest™， 最高质量的matlab作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此matlab作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Topological Properties of Metric Spaces

Let $X=(X, d)$ be a metric space. Defining, for every $x \in X$, the family $\mathcal{B}_x$ of neighborhoods of $x$ as the family of open balls centered at $x$
$$\mathcal{B}_x={B(x, \varepsilon), \varepsilon>0}$$
we introduce in $X$ a topology induced by the metric $d$. Thus every metric space is a topological space with the topology induced by the metric. Two immediate corollaries follow:
(i) Bases $\mathcal{B}_x$ are of countable type.
(ii) The metric topology is Hausdorff.
The first observation follows from the fact that $\mathcal{B}_x$ is equivalent to its subbase of the form
$$\left{B\left(x, \frac{1}{k}\right), \quad k=1,2, \ldots\right}$$
To prove the second assertion consider two distinct points $x \neq y$. We claim that balls $B(x, \varepsilon)$ and $B(y, \varepsilon)$, where $\varepsilon=d(x, y) / 2$, are disjoint. Indeed, if $z$ were a point belonging to the balls simultaneously, then
$$d(x, y) \leq d(x, z)+d(z, y)<\varepsilon+\varepsilon=d(x, y)$$
Thus all the results we have derived in the first five sections of this chapter for Hausdorff first countable topological spaces hold also for metric spaces. Let us briefly review some of them.

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Open and Closed Sets in Metric Spaces

Open and Closed Sets in Metric Spaces. A set $G \subset X$ is open if and only if, for every point $x$ of $G$, there exists a ball $B(x, \varepsilon)$, centered at $x$, that is contained in $G$. A point $x$ is an accumulation point of a set $F$ if every ball centered at $x$ contains points from $F$ which are different from $x$, or, equivalently, there exists a sequence $x_n$ points of $F$ converging to $x$.
Note that a sequence $x_n$ converges to $x$ if and only if
$$\forall \varepsilon>0 \exists N=N(\varepsilon): d\left(x_n, x\right)<\varepsilon \quad \forall n \geq N$$
Finally, a set is closed if it contains all its accumulation points.
Continuity in Metric Spaces. Let $(X, d)$ and $(Y, \rho)$ be two metric spaces. Recall that a function $f: X \rightarrow$ $Y$ is continuous at $x_0$ if
$$f\left(\mathcal{B}{x_0}\right) \succ \mathcal{B}{f\left(x_0\right)}$$
or, equivalently,
$$\forall \varepsilon>0 \quad \exists \delta>0: f\left(B\left(x_0, \delta\right)\right) \subset B\left(f\left(x_0\right), \varepsilon\right)$$
The last condition can be put into a more familiar form of the definition of continuity for metric spaces $(\varepsilon-\delta$ continuity):
Function $f: X \rightarrow Y$ is continuous at $x_0$ if and only if for every $\varepsilon>0$ there is a $\delta=\delta\left(\varepsilon, x_0\right)$ such that
$$\rho\left(f(x), f\left(x_0\right)\right)<\varepsilon \quad \text { whenever } \quad d\left(x, x_0\right)<\delta$$ Note that number $\delta$ generally depends not only on $\varepsilon$, but also upon the choice of point $x_0$. If $\delta$ happens to be independent of $x_0$ for all $x_0$ from a set $E$, then $f$ is said to be uniformly continuous on $E$. Let us recall also that, since bases of neighborhoods are of countable type, i.e., metric spaces are first countable topological spaces, continuity in metric spaces is equivalent to sequential continuity: a function $f: X \rightarrow Y$ is continuous at $x_0$ if and only if $$f\left(x_n\right) \rightarrow f\left(x_0\right) \quad \text { whenever } \quad x_n \rightarrow x_0$$ Suppose now that there exists a constant $C>0$, such that
$$\rho(f(x), f(y)) \leq C d(x, y) \quad \text { for every } x, y \in E$$

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Topological Properties of Metric Spaces

$$\mathcal{B}_x={B(x, \varepsilon), \varepsilon>0}$$

(i)基数$\mathcal{B}_x$为可数型。
(ii)度量拓扑是Hausdorff。

$$\left{B\left(x, \frac{1}{k}\right), \quad k=1,2, \ldots\right}$$

$$d(x, y) \leq d(x, z)+d(z, y)<\varepsilon+\varepsilon=d(x, y)$$

## 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。