数学代写|泛函分析作业代写Functional Analysis代考|MATH3051

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

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|泛函分析作业代写Functional Analysis代考|Basic Properties

DEFINITION 3.1 An operator $T$ from a normed space $V$ into another normed space $W$ is continuous at a point $x \in D(T)$ if for any $\epsilon>0$ there is a $\delta>0$, such that $|T x-T y|<\epsilon$ for all $y \in D(T)$ with $|x-y|<\delta . T$ is continuous if it is continuous at all points of $D(T)$. $T$ is uniformly continuous on $D(T)$ if for any $\epsilon>0$ there is a $\delta>0$, depending only on $\epsilon$, such that $|T x-T y|<\epsilon$ whenever $|x-y|<\delta$.
$T$ is bounded if there is a positive constant $M$ such that $|T x| \leq M|x|$ for all $x \in D(T)$

This is the obvious extension of the concept of continuity from real analysis, saying that objects close together are mapped into objects close to-gether. Continuous real functions in the usual sense are continuous operators in the sense above. Notice how the norm symbol $|\cdot|$ is used, denoting both the $V$-norm and the $W$-norm. In applications it is important to keep in mind in which space one is working, but it is a significant power of abstract mathematics to be able to write simple formulas and definitions.
Now we will show that continuity for linear operators is equivalent to boundedness.

数学代写|泛函分析作业代写Functional Analysis代考|Bounded Linear Operators

We will now return to the study of linear operators. Let $V$ and $W$ be two normed spaces, and let $S$ and $T$ be linear, bounded, and hence continuous operators from $V$ into $W$. It is obvious that for $\alpha, \beta \in C$, the operator $\alpha S+\beta T$ is also bounded (since it is continuous), so the bounded operators from $V$ into $W$ form a vector space. This space we will denote $B(V, W)$. In the special case where $V=W$, we will write $B(V)$ instead of $B(V, V)$. We will now equip $B(V, W)$ with a norm, the so-called operator norm.

DEFINITION 3.3 Let $V$ and $W$ be normed vector spaces. For $T \in$ $B(V, W)$, we define the norm of $T$ by
$$|T|=\sup {|T x| \mid|x| \leq 1} .$$
Notice how the norm symbol $|\cdot|$ is used for the norm in three different spaces. It is easy to see that $B(V, W)$ with the operator norm in fact is a normed space.

We observe that $|T x| \leq|T||x|$ for all $x \in V$, and that any $M>0$ satisfying $|T x| \leq M|x|$ for all $x \in V$ must also be $\geq|T|$.

Now, since the bounded operators form a vector space, we can consider sequences and series of bounded operators, and an important result is the following theorem.

数学代写|泛函分析作业代写Functional Analysis代考|Basic Properties

$T$ 如果有一个正常数是有界的 $M$ 这样 $|T x| \leq M|x|$ 对所有人 $x \in D(T)$

数学代写|泛函分析作业代写Functional Analysis代考|Bounded Linear Operators

$$|T|=\sup |T x||| x \mid \leq 1 .$$

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|泛函分析作业代写Functional Analysis代考|KMA322

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

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|泛函分析作业代写Functional Analysis代考|Infinite Dimensional Spaces

The Minkowski inequality shows, among other things, that the $L^{p}(R)$ spaces are in fact vector spaces, since they are stable under the vector space operations. This justifies the terminology we have been using from the very beginning. But since they differ quite a lot from the well-known finite dimensional ones, we will take a glance at how the basic definitions look.

Let $x_{1}, x_{2}, \ldots, x_{k}$ be elements in the vector space $V$. Recall that this set of vectors is called linearly independent if the equation
$$\alpha_{1} x_{1}+\alpha_{2} x_{2}+\ldots+\alpha_{k} x_{k}=0$$
is only satisfied for $\alpha_{1}=\alpha_{2}=\ldots=\alpha_{k}=0$. If this is not the case, the set is called linearly dependent. Notice also that any subset of a linearly independent set of vectors is again linearly independent. Moreover, a vector $x \in V$ is called a linear combination of the vectors $x_{1}, x_{2}, \ldots, x_{k}$ if there are constants $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{k}$, such that $\alpha_{1} x_{1}+\alpha_{2} x_{2}+\ldots+\alpha_{k} x_{k}=x$.

Let $U$ be a subset of $V$. By $\operatorname{span}(U)$ we denote the vector space consisting of all finite linear combinations of vectors in $U$. This is of course a subspace of $V$. If the vector space $V$ contains a finite set of $k$ linearly independent vectors $x_{1}, x_{2}, \ldots, x_{k}$, and moreover any set of $k+1$ vectors is linearly dependent, we will say that the dimension of $V$ is $k$, and write $\operatorname{dim}(V)=k$. It follows that any vector $x \in V$ is then a linear combination $x=\alpha_{1} x_{1}+\alpha_{2} x_{2}+\ldots+\alpha_{k} x_{k}$ of at most $k$ linearly independent vectors. If there is no such $k \in N$, we will say that $V$ is infinite dimensional.

数学代写|泛函分析作业代写Functional Analysis代考|Bounded Operators

A mapping from a normed space $V$ into another normed space $W$ is called an operator. If the mapping is linear, that is, if
$$T(\alpha x+\beta y)=\alpha T x+\beta T y,$$
for all $\alpha, \beta$ in $C$ and $x, y$ in $V$, we call $T$ a linear operator. We will sometimes consider operators defined only on a subset of $V$, called the domain of $T$ and usually denoted $D(T)$. If so, in order for $T$ to be linear, we will also demand that $D(T)$ is a subspace of $V$. In linear algebra we have studied the linear operators from $C^{n}$ into $C^{k}$, the matrices. The aim is now to extend the setting to study continuity properties of operators when the spaces are only assumed to be normed. Since normed spaces are topological spaces with the topology defined by the norms, the definition of continuity can now be written in a perhaps more intuitive manner.

数学代写|泛函分析作业代写Functional Analysis代考|Infinite Dimensional Spaces

Minkowski 不等式表明，除其他外， $L^{p}(R)$ 空间实际上是向量空间，因为它们在向量空间操作下是稳定的。这证 明了我们从一开始就使用的术语。但是由于它们与众所周知的有限维的有很大不同，我们将看一下基本定义的外 观。

$$\alpha_{1} x_{1}+\alpha_{2} x_{2}+\ldots+\alpha_{k} x_{k}=0$$

$x=\alpha_{1} x_{1}+\alpha_{2} x_{2}+\ldots+\alpha_{k} x_{k}$ 最多的 $k$ 线性独立向量。如果没有这样的 $k \in N$ ，我们会说 $V$ 是无限维的。

数学代写|泛函分析作业代写Functional Analysis代考|Bounded Operators

$$T(\alpha x+\beta y)=\alpha T x+\beta T y,$$

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|泛函分析作业代写Functional Analysis代考|MATH3402

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

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|泛函分析作业代写Functional Analysis代考|Metric Spaces

The study of continuous mappings between topological spaces is denoted general topology, and is a rich and vital area of mathematics. We will, however, in this context usually demand that the spaces we work in have even more structure. We would like the space to have a natural way to measure distances between elements, giving rise to the notion of metric spaces.

DEFINITION 1.2 Let $M$ be a set and $d: M \times M \rightarrow[0 ; \infty[$ a function. Then $(M, d)$ is denoted a metric space and $d$ a metric if for all $x, y, z$ in $M$ we have:
(i) $0 \leq d(x, y)$
(ii) $d(x, y)=0$ if and only if $x=y$
(iii) $d(x, y)=d(y, x)$
(iv) $d(x, y) \leq d(x, z)+d(z, y)$.
The condition (iv) above is denoted the triangle inequality.

数学代写|泛函分析作业代写Functional Analysis代考|Normed Vector Spaces

DEFINITION $2.1$ Let $V$ be a vector space. A norm on $V$ is a function $|\cdot| \rightarrow[0 ; \infty[$ satisfying for all $x, y \in V$ and $\alpha \in C$ :
\begin{aligned} |x+y| & \leq|x|+|y| \ |\alpha x| &=\mid \alpha|x| \ |x| &=0 \Rightarrow x=0 \end{aligned}
If the last condition above is omitted from the definition, $|\cdot|$ is called a semi norm.

We notice that any normed vector space is a metric space when we use the norm induced metric $d(x, y)=|\dot{x}-y|$. Notice also that any norm is continuous, since ||$x|-| y|\mid \leq| x-y |$.

DEFINITION 2.2 A normed vector space that is complete in the metric induced by the norm is called a Banach space.

数学代写|泛函分析作业代写Functional Analysis代考|Metric Spaces

(二) $d(x, y)=0$ 当且仅当 $x=y$
$\Leftrightarrow d(x, y)=d(y, x)$
(四) $d(x, y) \leq d(x, z)+d(z, y)$.

数学代写|泛函分析作业代写Functional Analysis代考|Normed Vector Spaces

$$|x+y| \leq|x|+|y||\alpha x| \quad=|\alpha| x|| x \mid=0 \Rightarrow x=0$$

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。