数学代写|偏微分方程代写partial difference equations代考|MAP4341

2023年2月2日

couryes-lab™ 为您的留学生涯保驾护航 在代写偏微分方程partial difference equations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。我们的专家在代写偏微分方程partial difference equations代写方面经验极为丰富，各种代写偏微分方程partial difference equations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
couryes™为您提供可以保分的包课服务

数学代写|偏微分方程代写partial difference equations代考|Continuous and compact operators

A linear mapping $T$ is continuous if and only if the image of the unit sphere under $T$ is bounded (see Appendix A.1). If this image is even relatively compact (that is, its closure is compact), then we call $T$ compact. We have thus made the following definition:

Definition 4.37 Let $E$ and $F$ be normed spaces. A linear mapping $T: E \rightarrow F$ is called compact if the following condition is satisfied: for any bounded sequence $\left(u_n\right){n \in \mathbb{N}} \subset E$ there exists a subsequence $\left(u{n_k}\right){k \in \mathbb{N}}$ such that $\left(T u{n_k}\right){k \in \mathbb{N}}$ converges in $F{\text {s }}$

Here we wish to consider such mappings in the special situation of Hilbert spaces, where we have both convergence in norm and weak convergence. We will assume throughout that $H_1$ and $H_2$ are real or complex separable Hilbert spaces.

Theorem 4.38 Let $T: H_1 \rightarrow H_2$ be linear and continuous. Then $T$ is weakly continuous, that is, $u_n \rightarrow u$ in $H_1$ implies $T u_n \rightarrow T u$ in $H_2$.

Proof Let $w \in H_2$, then $\varphi(v):=(T v, w){H_2}$ defines a continuous linear form on $H_1$. By the theorem of Riesz-Fréchet, there is thus a unique $T^* w \in H_1$ such that $(T v, w){H_2}=\left(v, T^* w\right){H_1}$ for all $v \in H_1$. If $u_n \rightarrow u$ in $H_1$, then $$\left(T u_n, w\right){H_2}=\left(u_n, T^* w\right){H_1} \rightarrow\left(u, T^* w\right){H_1}=(T u, w)_{H_2} .$$
Since $w \in \mathrm{H}_2$ was arbitrary, it follows that $T u_n \rightarrow T u$ in $H_2$.
The converse of Theorem $4.38$ is also true: every weakly continuous operator is also continuous; see Exercise 4.2. Now we can describe compact operators as being exactly those operators which “improve” convergence in the following sense.

Theorem 4.39 Let $T: H_1 \rightarrow H_2$ be linear. Then the following statements are equivalent:
(i) $T$ is compact.
(ii) $u_n \rightarrow u$ in $H_1$ implies $T u_n \rightarrow T u$ in $\mathrm{H}_2$.
For the proof we will use the following lemma which, despite looking like splitting hairs, turns out to be extremely useful.

David Hilbert (1862-1943) was one of the pre-eminent mathematicians of the first half of last century. At the International Congress of Mathematicians (ICM) in Paris in 1900, Hilbert presented his famous 23 mathematical problems, which continue to exert considerable influence on mathematical research to this day. He was the central figure of the Göttingen school, which up until the Nazis seized power in Germany in 1933 was one of the foremost mathematical institutions worldwide. Between 1904 and 1910, Hilbert published a series of articles on integral equations, in which bilinear forms on $\ell_2$ appeared as an essential tool.

The definition of Hilbert spaces was only given in 1929, by John von Neumann. Decisive for the success of Hilbert space theory was the invention of the Lebesgue integral, in the doctoral thesis of Henri Lebesgue completed in 1902. Building on Lebesgue’s theory, Frigyes Riesz showed in 1907 that every element of $\ell_2(\mathbb{Z})$ is the sequence of Fourier coefficients of some function in $L_2(0,2 \pi)$. In the same year, Ernst Fischer (1875-1954) proved that $L_2(0,2 \pi)$ is complete, which is equivalent to Riesz’ result. This is why the name “theorem of Riesz-Fischer” is often used for the fact that $L_2(\Omega)$

is complete (see the appendix). Riesz and Maurice Fréchet (1878-1973) determined the set of all continuous linear forms on $L_2(0,1)$ independently of each other in 1907, thus proving Theorem $4.21$.

Fréchet obtained his doctorate in Paris in 1906 under Hadamard; in his dissertation, metric spaces are introduced for the first time. The theorem of Riesz-Fréchet is often known as the Riesz representation theorem; however, there is another Riesz representation theorem for measures, which will play an important role in Section 7.2.

It was Hilbert who proved a first version of the spectral theorem, Theorem 4.50. But subsequent to his works on integral equations, it was operators and not bilinear forms which took centre stage. Bounded operators on Hilbert spaces became a central concept in mathematics, and to this day operator theory remains an important area of mathematical research. In the 1930 s. quantum theory was a key driving force in the development of functional analysis. The decisive breakthrough in the mathematieal formulation of quantum physies was made by John von Neumann. who had visited Göttingen in 1926/27 and to whom the concept of unbounded operators is due (cf. Mathematische Annalen, No. 33, 1932). An unbounded self-adjoint operator (as was defined in Section 4.8) models an observable in quantum theory. The book Mathematische Grundlagen der Quantenmechanik (Mathematical Foundations of Quantum Mechanics) by John von Neumann, which appeared in 1932 in German, describes the mathematical modeling of quantum theory as is still used today, and indeed with great success. Although operators are ideal for describing equations, it is interesting that bilinear forms reached their pinnacle as a method for solving partial differential equations around 50 years after Hilbert’s work.

偏微分方程代写

数学代写|偏微分方程代写partial difference equations代考|Continuous and compact operators

$$\left(T u_n, w\right) H_2=\left(u_n, T^* w\right) H_1 \rightarrow\left(u, T^* w\right) H_1$$

(i) $T$ 很紧湊。
(二) $u_n \rightarrow u$ 在 $H_1$ 暗示 $T u_n \rightarrow T u$ 在 $\mathrm{H}_2$.

Fréchet 于 1906 年在 Hadamard 的指导下在巴黎获得 博士学位。在他的论文中，首次引入了度量空间。
Riesz-Fréchet 定理通常被称为 Riesz 表示定理；然而， 还有另一个测度的 Riesz 表示定理，它将在 $7.2$ 节中发 挥重要作用。

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。