## 数学代写|泛函分析作业代写Functional Analysis代考|MAT4450

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。

## 数学代写|泛函分析作业代写Functional Analysis代考|Generalization for Closed Operators

Surprising as it looks, most of the results from the preceding two sections can be generalized to the case of closed operators.

Topological Transpose. Let $X$ and $Y$ be two normed spaces and let $A: X \supset D(A) \rightarrow Y$ be a linear operator, not necessarily continuous. Consider all points $\left(\boldsymbol{y}^{\prime}, \boldsymbol{x}^{\prime}\right)$ from the product space $Y^{\prime} \times X^{\prime}$ such that
$$\left\langle\boldsymbol{y}^{\prime}, A \boldsymbol{x}\right\rangle=\left\langle\boldsymbol{x}^{\prime}, \boldsymbol{x}\right\rangle \quad \forall \boldsymbol{x} \in D(A)$$
where the duality pairings are to be understood in $Y^{\prime} \times Y$ and $X^{\prime} \times X$, respectively. Notice that the set is nonempty as it always contains point $(\mathbf{0}, \mathbf{0})$. We claim that $\boldsymbol{y}^{\prime}$ uniquely defines $\boldsymbol{x}^{\prime}$ iff the domain $D(A)$ of operator $A$ is dense in $X$. Indeed, assume that $\overline{D(A)}=X$. By linearity of both sides with respect to the first argument, it is sufficient to prove that
$$\left\langle\boldsymbol{x}^{\prime}, \boldsymbol{x}\right\rangle=0 \quad \forall \boldsymbol{x} \in D(A) \quad \text { implies } \quad \boldsymbol{x}^{\prime}=\mathbf{0}$$
But this follows easily from the density of $D(A)$ in $X$ and continuity of $\boldsymbol{x}^{\prime}$.
Conversely, assume that $\overline{D(A)} \neq X$. Let $x \in X-\overline{D(A)}$. By the Mazur Separation Theorem (Lemma 5.13.1) there exists a continuous and linear functional $\boldsymbol{x}_0^{\prime}$, vanishing on $\overline{D(A)}$, but different from zero at $\boldsymbol{x}$. Consequently, the zero functional $\boldsymbol{y}^{\prime}=\mathbf{0}$ has two corresponding elements $\boldsymbol{x}^{\prime}=\mathbf{0}$ and $\boldsymbol{x}^{\prime}=\boldsymbol{x}_0^{\prime}$, a contradiction.

Thus, restricting ourselves to the case of operators $A$ defined on domains $D(A)$ which are dense in $X$, we can identify the collection of $\left(\boldsymbol{y}^{\prime}, \boldsymbol{x}^{\prime}\right)$ discussed above (see Proposition 5.10.1) as the graph of a linear operator from $Y^{\prime}$ to $X^{\prime}$, denoted $A^{\prime}$, and called the transpose (or dual) of operator $A$. Due to our construction, this definition generalizes the definition of the transpose for $A \in \mathcal{L}(X, Y)$.

## 数学代写|泛函分析作业代写Functional Analysis代考|Closed Range Theorem for Closed Operators

As we have indicated in Section 5.18, we encounter some fundamental technical difficulties in extending the arguments used for the continuous operators. Our exposition follows the proof by Tosio Kato ([5], Theorem 4.8 and Section 5.2).

We start with Kato’s fundamental geometrical result on orthogonal components. Let $Z$ be a Banach space, and let $M, N$ denote two closed subspaces of $Z$. The intersection $M \cap N$ is obviosuly closed but, in the infinite dimensional setting, the direct sum $M \oplus N$ may not be closed.

The following property follows directly from the definition of the orthogonal complement (comp. Exercise 5.19 .1$)$.
$$(M+N)^{\perp}=M^{\perp} \cap N^{\perp}$$
The corresponding property for the orthogonal complement of $M \cap N$ is far from trivial.

(Kato’s Theorem)
Let $Z$ be a Banach space and $M, N$ be two closed subspaces of $Z$. Then subspace $M+N$ is closed in $Z$ if and only if subspace $M^{\perp}+N^{\perp}$ is closed in $Z^{\prime}$ and
$$M^{\perp}+N^{\perp}=(M \cap N)^{\perp}$$
LEMMA 5.19.1
Let $M+N$ be closed in Z. Then the identity (5.5) holds and, in particular, $M^{\perp}+N^{\perp}$ is closed in $Z^{\prime}$.

PROOF Step 1. Assume additionally that $M \cap N={\mathbf{0}}$. Obviously, ${\mathbf{0}}^{\perp}=\mathbf{Z}^{\prime}$, so we need to prove that
$$M^{\perp}+N^{\perp}=Z^{\prime}$$
Inclusion $\subset$ is trivial. To prove inclusion $\supset$, take an arbitrary $f \in Z^{\prime}$. Let $z \in M+N$ and
$$\boldsymbol{z}=\boldsymbol{m}+\boldsymbol{n}, \quad \boldsymbol{m} \in M, \boldsymbol{n} \in N$$
be the unique decomposition of $\boldsymbol{z}$. Consider linear projections implied by the decomposition,
$$\begin{gathered} P_M: M+N \ni \boldsymbol{z} \rightarrow \boldsymbol{m} \in M \ P_N: M+N \ni \boldsymbol{z} \rightarrow \boldsymbol{n} \in N \end{gathered}$$

# 泛函分析代写

## 数学代写|泛函分析作业代写Functional Analysis代考|Generalization for Closed Operators

$$\left\langle\boldsymbol{y}^{\prime}, A \boldsymbol{x}\right\rangle=\left\langle\boldsymbol{x}^{\prime}, \boldsymbol{x}\right\rangle \quad \forall \boldsymbol{x} \in D(A)$$

$$\left\langle\boldsymbol{x}^{\prime}, \boldsymbol{x}\right\rangle=0 \quad \forall \boldsymbol{x} \in D(A) \quad \text { implies } \quad \boldsymbol{x}^{\prime}=\mathbf{0}$$

## 数学代写|泛函分析作业代写Functional Analysis代考|Closed Range Theorem for Closed Operators

$$(M+N)^{\perp}=M^{\perp} \cap N^{\perp}$$
$M \cap N$的正交补的相应性质远非平凡。

(加藤定理)

$$M^{\perp}+N^{\perp}=(M \cap N)^{\perp}$$

$$M^{\perp}+N^{\perp}=Z^{\prime}$$

$$\boldsymbol{z}=\boldsymbol{m}+\boldsymbol{n}, \quad \boldsymbol{m} \in M, \boldsymbol{n} \in N$$

$$\begin{gathered} P_M: M+N \ni \boldsymbol{z} \rightarrow \boldsymbol{m} \in M \ P_N: M+N \ni \boldsymbol{z} \rightarrow \boldsymbol{n} \in N \end{gathered}$$

## 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|拓扑学代写Topology代考|Darboux’s theorem

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

## 数学代写|拓扑学代写Topology代考|Darboux’s theorem

Theorem 2.5.1 (Darboux’s theorem) Let $\alpha$ be a contact form on the $(2 n+1)$-dimensional manifold $M$ and $p$ a point on $M$. Then there are coordinates $x_1, \ldots, x_n, y_1, \ldots, y_n, z$ on a neighbourhood $U \subset M$ of $p$ such that $p=(0, \ldots, 0)$ and
$$\left.\alpha\right|U=d z+\sum{j=1}^n x_j d y_j .$$
Remark 2.5.2 Observe that the $\operatorname{map}(\mathbf{x}, \mathbf{y}, z) \mapsto\left(\varepsilon \mathbf{x}, \varepsilon \mathbf{y}, \varepsilon^2 z\right)$ is a contactomorphism of the standard contact structure $\xi_{\text {st }}$ on $\mathbb{R}^{2 n+1}$ for any $\varepsilon \in \mathbb{R}^{+}$. Therefore it is an immediate consequence of the Darboux theorem that there is a contact embedding of the closed unit ball $B_{\mathrm{st}}$ in $\left(\mathbb{R}^{2 n+1}, \xi_{\mathrm{st}}\right)$ into $(M, \xi=\operatorname{ker} \alpha)$ which sends the origin to $p$. Here ‘contact embedding of $B_{\mathrm{st}}$ ‘ simply means a contactomorphism of a small open neighbourhood of $B_{\mathrm{st}}$ in $\left(\mathbb{R}^{2 n+1}, \xi_{\text {st }}\right)$ onto its image in $(M, \xi)$; later we shall encounter a more general concept of contact embeddings.

In fact, by Proposition 2.1.8 and Example 2.1.10 there is a contactomorphism of $\left(\mathbb{R}^{2 n+1}, \xi_{\mathrm{st}}\right)$ with a relatively compact subset of itself, and hence by scaling with a subset of $B_{\text {st }}$. So we can also construct a contactomorphism between $\left(\mathbb{R}^{2 n+1}, \xi_{\text {st }}\right)$ and a neighbourhood of $p$ in $(M, \xi)$.

Proof of Theorem 2.5.1 We may assume without loss of generality that $M=\mathbb{R}^{2 n+1}$ and $p=\mathbf{0}$ is the origin of $\mathbb{R}^{2 n+1}$. Choose linear coordinates
$$x_1, \ldots, x_n, y_1, \ldots y_n, z$$
on $\mathbb{R}^{2 n+1}$ such that
$$\text { on } T_0 \mathbb{R}^{2 n+1}:\left{\begin{array}{l} \alpha\left(\partial_z\right)=1, \quad i_{\partial_z} d \alpha=0, \ \partial_{x_j}, \partial_{y_j} \in \operatorname{ker} \alpha(j=1, \ldots, n), d \alpha=\sum_{j=1}^n d x_j \wedge d y_j . \end{array}\right.$$

## 数学代写|拓扑学代写Topology代考|Isotropic submanifolds

Let $L \subset(M, \xi=\operatorname{ker} \alpha)$ be an isotropic submanifold in a contact manifold with cooriented contact structure. Write $\left.(T L)^{\perp} \subset \xi\right|_L$ for the sub-bundle of $\left.\xi\right|L$ that is symplectically orthogonal to $T L$ with respect to the symplectic bundle structure $\left.d \alpha\right|{\xi}$. As we have seen in the preceding symplectic interlude, the conformal class of this symplectic bundle structure only depends on the contact structure $\xi$, not on the choice of contact form $\alpha$ defining $\xi$. So the bundle $(T L)^{\perp}$ is determined by $\xi$.

The fact that $L$ is isotropic implies $T L \subset(T L)^{\perp}$. Lemma 1.3 .3 allows us to make the following definition, see [241].
Definition 2.5.3 The quotient bundle
$$\operatorname{CSN}_M(L):=(T L)^{\perp} / T L$$
with the conformal symplectic structure induced by $d \alpha$ is called the conformal symplectic normal bundle of $L$ in $M$.
So the normal bundle $N L:=\left(\left.T M\right|_L\right) / T L$ of $L$ in $M$ can be split as
$$N L \cong\left(\left.T M\right|_L\right) /\left(\left.\xi\right|_L\right) \oplus\left(\left.\xi\right|_L\right) /(T L)^{\perp} \oplus \operatorname{CSN}_M(L) .$$

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Darboux’s theorem

$$\left.\alpha\right|U=d z+\sum{j=1}^n x_j d y_j .$$
2.5.2注意:对于任何$\varepsilon \in \mathbb{R}^{+}$, $\operatorname{map}(\mathbf{x}, \mathbf{y}, z) \mapsto\left(\varepsilon \mathbf{x}, \varepsilon \mathbf{y}, \varepsilon^2 z\right)$都是$\mathbb{R}^{2 n+1}$上的标准触点结构$\xi_{\text {st }}$的触点形态。因此，达布定理的直接结果是，$\left(\mathbb{R}^{2 n+1}, \xi_{\mathrm{st}}\right)$中的闭合单位球$B_{\mathrm{st}}$在$(M, \xi=\operatorname{ker} \alpha)$中有一个接触嵌入，它将原点发送到$p$。在这里，“$B_{\mathrm{st}}$的接触嵌入”仅仅意味着将$\left(\mathbb{R}^{2 n+1}, \xi_{\text {st }}\right)$中的一个小的开放邻域$B_{\mathrm{st}}$与其在$(M, \xi)$中的图像的接触形态;稍后我们将遇到更一般的接触嵌入概念。

$$x_1, \ldots, x_n, y_1, \ldots y_n, z$$

$$\text { on } T_0 \mathbb{R}^{2 n+1}:\left{\begin{array}{l} \alpha\left(\partial_z\right)=1, \quad i_{\partial_z} d \alpha=0, \ \partial_{x_j}, \partial_{y_j} \in \operatorname{ker} \alpha(j=1, \ldots, n), d \alpha=\sum_{j=1}^n d x_j \wedge d y_j . \end{array}\right.$$

## 数学代写|拓扑学代写Topology代考|Isotropic submanifolds

$L$是各向同性的这一事实意味着$T L \subset(T L)^{\perp}$。引理1.3 .3允许我们做出如下定义，参见[241]。
2.5.3商束
$$\operatorname{CSN}_M(L):=(T L)^{\perp} / T L$$

$$N L \cong\left(\left.T M\right|_L\right) /\left(\left.\xi\right|_L\right) \oplus\left(\left.\xi\right|_L\right) /(T L)^{\perp} \oplus \operatorname{CSN}_M(L) .$$

## 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|拓扑学代写Topology代考|Cerf ’s theorem

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

## 数学代写|拓扑学代写Topology代考|Cerf ’s theorem

Write $\operatorname{Diff}(M)$ for the group of orientation-preserving diffeomorphisms of an orientable differential manifold $M$ (the group multiplication being given by composition of diffeomorphisms). Let $D^n$ be the $n$-dimensional unit disc in $\mathbb{R}^n$, and $S^{n-1}=\partial D^n$ its boundary, the standard $(n-1)$-dimensional unit sphere. Since diffeomorphisms of a manifold with boundary preserve that boundary, we have a natural restriction homomorphism
$$\begin{array}{ccc} \rho_n: \quad \operatorname{Diff}\left(D^n\right) & \longrightarrow \operatorname{Diff}\left(S^{n-1}\right) \ f & \longmapsto & \left.f\right|_{S^{n-1}} . \end{array}$$
The group $\Gamma_n$ is defined as
$$\Gamma_n=\operatorname{Diff}\left(S^{n-1}\right) / \operatorname{im} \rho_n .$$
In order to show that this is indeed a group, we need to prove that $\operatorname{im} \rho_n$ is a normal subgroup in $\operatorname{Diff}\left(S^{n-1}\right)$.

We begin with two lemmata. Write $\operatorname{Diff}_0\left(S^{n-1}\right)$ for the group of diffeomorphisms of $S^{n-1}$ that are isotopic to the identity.

## 数学代写|拓扑学代写Topology代考|Property P for knots

We begin by recalling a few facts about Dehn surgery on knots in 3 -manifolds, mostly to set up notation. For a textbook reference on this topic see [209] or [215].

Let $K$ be a knot in the 3 -sphere $S^3$ (or, more generally, in some oriented 3 manifold $M$ ), by which we mean a smoothly embedded copy of the circle $S^1$. Write $\nu K$ for a (closed) tubular neighbourhood of $K$. The neighbourhood $\nu K$ is diffeomorphic to a solid torus $S^1 \times D^2$, since this is the only orientable $D^2$-bundle over $S^1$. Let $C$ be the closure of the complement $S^3 \backslash \nu K$ of $\nu K$ in $S^3$. (Part of) the Mayer-Vietoris sequence $\dagger$ for $S^3=\nu K \cup C$ with $\nu K \cap C=T^2$ reads
\begin{aligned} & H_2\left(S^3\right) \rightarrow H_1\left(T^2\right) \quad \rightarrow \quad H_1(\nu K) \oplus H_1(C) \rightarrow H_1\left(S^3\right) \ & 0 \quad \rightarrow \mathbb{Z} \oplus \mathbb{Z} \rightarrow \mathbb{Z} \quad \oplus \quad H_1(C) \rightarrow 0 \quad 0 . \ & \end{aligned}
We conclude that $H_1(C) \cong \mathbb{Z}$. We also see that on $T^2=\partial(\nu K)$ there are two distinguished curves, unique up to isotopy.
(1) The meridian $\mu$, defined as a simple closed curve that generates the kernel of the homomorphism $H_1\left(T^2\right) \rightarrow H_1(\nu K)$.
(2) The preferred longitude $\lambda$, a simple closed curve that generates the kernel of the homomorphism $H_1\left(T^2\right) \rightarrow H_1(C)$.

We assume that $S^3$ is equipped with its standard orientation as the boundary of $D^4 \cdot \ddagger$ We give $T^2=\partial(\nu K)$ the boundary orientation. We also assume $K$ to be oriented. Then $\lambda$ can be oriented by requiring it to be isotopic to $K$ in $\nu K$ as oriented curve; the orientation we choose for $\mu$ is the one that turns $\mu, \lambda$ into a positive basis for that homology group. (Occasionally we allow ourselves to denote a simple closed curve on $T^2$ by the same symbol as the class it represents in $H_1\left(T^2\right)$, since that class determines the curve up to isotopy.) This is illustrated in Figure 1.5, with the standard (right-hand) orientation of ambient 3-space.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Cerf ’s theorem

$$\begin{array}{ccc} \rho_n: \quad \operatorname{Diff}\left(D^n\right) & \longrightarrow \operatorname{Diff}\left(S^{n-1}\right) \ f & \longmapsto & \left.f\right|_{S^{n-1}} . \end{array}$$

$$\Gamma_n=\operatorname{Diff}\left(S^{n-1}\right) / \operatorname{im} \rho_n .$$

## 数学代写|拓扑学代写Topology代考|Property P for knots

\begin{aligned} & H_2\left(S^3\right) \rightarrow H_1\left(T^2\right) \quad \rightarrow \quad H_1(\nu K) \oplus H_1(C) \rightarrow H_1\left(S^3\right) \ & 0 \quad \rightarrow \mathbb{Z} \oplus \mathbb{Z} \rightarrow \mathbb{Z} \quad \oplus \quad H_1(C) \rightarrow 0 \quad 0 . \ & \end{aligned}

(1)子午线$\mu$，定义为生成同态核$H_1\left(T^2\right) \rightarrow H_1(\nu K)$的简单封闭曲线。
(2)首选经度$\lambda$，一条生成同态核$H_1\left(T^2\right) \rightarrow H_1(C)$的简单封闭曲线。

## 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|拓扑学代写Topology代考|Universal Coverings

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

## 数学代写|拓扑学代写Topology代考|Universal Coverings

Definition 13.29 A covering $u: \widetilde{X} \rightarrow X$ is said to be universal if the total space $\widetilde{X}$ is connected and simply connected.

We saw in Example 13.22 that universal coverings are regular. In particular, if $u: \widetilde{X} \rightarrow X$ is universal then $\operatorname{Aut}(\widetilde{X}, u)$ acts freely and transitively on fibres, and $\pi_1(X) \simeq \operatorname{Aut}(\widetilde{X}, u)$ are isomorphic.

Proposition 13.30 (Universal property of universal coverings) Let $u: \widetilde{X} \rightarrow X$ be a universal covering. For every covering $p: E \rightarrow X$ and any points $\tilde{x} \in \tilde{X}, e \in E$ such that $u(\tilde{x})=p(e)$, there exists a unique covering morphism $\phi: \widetilde{X} \rightarrow E$ such that $\phi(\tilde{x})=e$. In particular, all universal coverings of a space $X$ are isomorphic to one another.

Proof Since $0=u_* \pi_1(\tilde{X}, \tilde{x}) \subset p_* \pi_1(E, e), \phi$ exists by virtue of Theorem 13.18. Additionally, if $p: E \rightarrow X$ is universal then the previous arguments show that there’s a covering morphism $\psi: E \rightarrow \widetilde{X}$ with $\psi(e)=\tilde{x}$. But $\widetilde{X}$ and $E$ are connected by definition, so the lift’s uniqueness forces $\phi \psi$ and $\psi \phi$ to be identity maps.

This proves the uniqueness of universal coverings. The remaining part of the section is devoted entirely to the issue of existence. We begin with a simple necessary condition.

## 数学代写|拓扑学代写Topology代考|Coverings with Given Monodromy

Consider a covering $p: E \rightarrow X$, a point $x \in X$ and the monodromy action
$$p^{-1}(x) \times \pi_1(X, x) \rightarrow p^{-1}(x) .$$
It’s not hard to show that $E$ is connected if and only if the monodromy action is transitive. In fact if $E$ is connected, for every pair $a, b \in p^{-1}(x)$ we can find a path $\alpha \in \Omega(E, a, b)$ and hence $b=a \cdot[p \alpha]$. Conversely, if the monodromy is transitive, the fibre $p^{-1}(x)$ is contained in a path component. Given any point $a \in E$ we choose a path $\alpha: I \rightarrow X$ such that $\alpha(0)=p(a), \alpha(1)=x$. The lift $\alpha_a: I \rightarrow E$ joins $a$ to some point in $p^{-1}(x)$, so that $a$ belongs in the same connected component where $p^{-1}(x)$ lies.

We saw already, in Theorem 13.1, that the stabiliser of any $e \in p^{-1}(x)$, i.e. the subgroup
$$\operatorname{Stab}(e)=\left{a \in \pi_1(X, x) \mid e \cdot a=e\right},$$
coincides with $p_* \pi_1(E, e)$. In particular the covering $p: E \rightarrow X$ is universal if and only if the monodromy is free and transitive. Moreover, the covering is regular if and only if the monodromy acts transitively and all stabilisers are normal subgroups.
Theorem 13.35 Let $X$ be connected, locally path connected and semi-locally simply connected. For every non-empty set $T$ and every right action
$$T \times \pi_1(X, x) \stackrel{\bullet}{\longrightarrow} T$$
there exists a covering $p: E \rightarrow X$ and a bijection $\phi: T \rightarrow p^{-1}(x)$ such that $\phi(t \bullet a)=\phi(t) \cdot a$, for every $t \in T$ and $a \in \pi_1(X, x)$. The pair $(p, \phi)$ is unique up to isomorphism.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Coverings with Given Monodromy

$$p^{-1}(x) \times \pi_1(X, x) \rightarrow p^{-1}(x) .$$

$$\operatorname{Stab}(e)=\left{a \in \pi_1(X, x) \mid e \cdot a=e\right},$$

$$T \times \pi_1(X, x) \stackrel{\bullet}{\longrightarrow} T$$

## 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|Example of a Closed Operator

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。

## 数学代写|泛函分析作业代写Functional Analysis代考|Example of a Closed Operator

Locally Integrable Functions. Let $\Omega \subset \mathbb{R}^n$ be an open set. A real- or complex-valued function $u$ defined on $\Omega$ is said to be locally integrable if, for every point $x \in \Omega$, there exists a ball $B=B(x, \epsilon) \subset \Omega$ such that the restriction of function $u$ to $B$ is summable in $B$, i.e., $\left.u\right|B \in L^1(B)$. Equivalently, for every compact set $K \subset \Omega,\left.u\right|_K \in L^1(K)$, comp. Exercise 5.11.1. The locally integrable functions form a vector space, denoted $L{l o c}^1(\Omega)$, that plays a crucial role in the theory of distributions.

Distributional Derivatives. Let $\Omega \subset \mathbb{R}^n$ be an open set, $\boldsymbol{\alpha}=\left(\alpha_1, \ldots, \alpha_n\right)$ a multi-index, and $u \in L^p(\Omega)$ an arbitrary $L^p$-function. A function $u^\alpha$ defined on $\Omega$ is called the distributional derivative of $u$, denoted $D^\alpha u$, iff
$$\int_{\Omega} u D^\alpha \varphi d x=(-1)^{|\alpha|} \int_{\Omega} u^\alpha \varphi d x \forall \varphi \in C_0^{\infty}(\Omega)$$
where $C_0^{\infty}(\Omega)$ is the space of test functions discussed in Section 5.3. (It is understood that function $u^\alpha$ must satisfy sufficient conditions for the right-hand side to exist.)

Notice that the notion of the distributional derivative is a generalization of the classical derivative. Indeed, in the case of a $C^{|\alpha|}$ function $u$, the formula above follows from the (multiple) integration by parts and the fact that test functions, along with their derivatives, vanish on the boundary $\partial \Omega$.

## 数学代写|泛函分析作业代写Functional Analysis代考|Examples of Dual Spaces, Representation Theorem for Topological Duals of $L^p$ Spaces

Let $f \in U^{\prime}=\mathcal{L}(U, \mathbb{R})$. As in Chapter 2, it is customary to represent the functional $f$ as a duality pairing; i.e., we usually write
$$f(\boldsymbol{u})=\langle f, \boldsymbol{u}\rangle, \quad f \in U^{\prime}, \quad \boldsymbol{u} \in U$$
Then the symbol $\langle\cdot, \cdot\rangle$ can be regarded as a bilinear map from $U^{\prime} \times U$ into $\mathbb{R}$ or $\mathbb{C}$.
Now, since $f(\boldsymbol{u})$ is a real or complex number, $|f(\boldsymbol{u})|=|\langle f, \boldsymbol{u}\rangle|$. Hence, in view of what was said about the norms on spaces $\mathcal{L}(U, V)$ of linear operators, the norm of an element of $U^{\prime}$ is given by
$$|f|_{U^{\prime}}=\sup {\boldsymbol{u} \in U}\left{\frac{|\langle f, \boldsymbol{u}\rangle|}{|\boldsymbol{u}|_U}, \boldsymbol{u} \neq \mathbf{0}\right}$$ Hence we always have $$|\langle f, \boldsymbol{u}\rangle| \leq|f|{U^{\prime}}|\boldsymbol{u}|_U \quad f \in U^{\prime}, \boldsymbol{u} \in U$$
which in particular implies that the duality pairing is continuous (explain, why?).
Before we proceed with some general results concerning dual spaces, we present in this section a few nontrivial examples of dual spaces in the form of so-called representation theorems. The task of a representation theorem is to identify elements from a dual space (i.e., linear and continuous functionals defined on a normed space) with elements from some other space, for instance some other functions, through a representation formula relating functionals with those functions. The representation theorems not only provide meaningful characterizations of dual spaces, but are also of great practical value in applications.

The main result we present in this chapter is the representation theorem for the duals of the spaces $L^p(\Omega)$, $1 \leq p<\infty$

# 泛函分析代写

## 数学代写|泛函分析作业代写Functional Analysis代考|Example of a Closed Operator

$$\int_{\Omega} u D^\alpha \varphi d x=(-1)^{|\alpha|} \int_{\Omega} u^\alpha \varphi d x \forall \varphi \in C_0^{\infty}(\Omega)$$

## 数学代写|泛函分析作业代写Functional Analysis代考|Examples of Dual Spaces, Representation Theorem for Topological Duals of $L^p$ Spaces

$$f(\boldsymbol{u})=\langle f, \boldsymbol{u}\rangle, \quad f \in U^{\prime}, \quad \boldsymbol{u} \in U$$

$$|f|_{U^{\prime}}=\sup {\boldsymbol{u} \in U}\left{\frac{|\langle f, \boldsymbol{u}\rangle|}{|\boldsymbol{u}|_U}, \boldsymbol{u} \neq \mathbf{0}\right}$$因此我们总是有$$|\langle f, \boldsymbol{u}\rangle| \leq|f|{U^{\prime}}|\boldsymbol{u}|_U \quad f \in U^{\prime}, \boldsymbol{u} \in U$$

## 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|Fundamental Properties of Linear Bounded Operators

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。

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

We now expand our study to linear transformations on normed spaces. Since the domains of such linear mappings now have topological structure, we can also apply many of the properties of functions on metric spaces. For example, we are now able to talk about continuous linear transformations from one normed linear space into another. It is not uncommon to use the term “operator” to refer to a mapping or function on sets that have both algebraic and topological structure. Since all of our subsequent work involves cases in which this is so, we henceforth use the term operator synonymously with function, mapping, and transformation.

To begin our study, let $\left(U,|\cdot|_U\right)$ and $\left(V,|\cdot|_V\right)$ denote two normed linear spaces over the same field $\mathbb{F}$, and let $A$ be an operator from $U$ into $V$. We recall that an operator $A$ from $U$ into $V$ is linear if and only if it is homogeneous (i.e., $A(\alpha \boldsymbol{u})=\alpha A \boldsymbol{u} \forall \boldsymbol{u} \in U$ and $\alpha \in \mathbb{F}$ ) and additive (i.e., $A\left(\boldsymbol{u}_1+\boldsymbol{u}_2\right)=$ $\left.A\left(\boldsymbol{u}_1\right)+A\left(\boldsymbol{u}_2\right) \forall \boldsymbol{u}_1, \boldsymbol{u}_2 \in U\right)$. Equivalently, $A: U \rightarrow V$ is linear if and only if $A\left(\alpha \boldsymbol{u}_1+\beta \boldsymbol{u}_2\right)=$ $\alpha A\left(\boldsymbol{u}_1\right)+\beta A\left(\boldsymbol{u}_2\right) \forall \boldsymbol{u}_1, \boldsymbol{u}_2 \in U$ and $\forall \alpha, \beta \in \mathbb{F}$. When $A$ does not obey this rule, it is called a nonlinear operator. In the sequel we shall always take the field $\mathbb{F}$ to be real or complex numbers: $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$.
Recall that the null space, $\mathcal{N}(A)$, of a linear operator $A: U \rightarrow V$ is defined by $\mathcal{N}(A)={\boldsymbol{u}: A u=$ $0, \boldsymbol{u} \in U}$ and is a subspace of $U$, and the range $\mathcal{R}(A)$ of a linear operator $A: U \rightarrow V$ is defined to be $\mathcal{R}(A)={v: A \boldsymbol{u}=\boldsymbol{v} \in V$, for $\boldsymbol{u} \in U}$ and $\mathcal{R}(A) \subset V$. We note here that the operator $A$ is one-to-one if and only if the null space $\mathcal{N}(A)$ is trivial, $\mathcal{N}(A)={0}$.

Thus far we have introduced only algebraic properties of linear operators. To talk about boundedness and continuity of linear operators, we use the topological structure of the normed spaces $U$ and $V$.

## 数学代写|泛函分析作业代写Functional Analysis代考|The Space of Continuous Linear Operators

In this section, we will more closely investigate the space $\mathcal{L}(U, V)$ of all continuous operators from a normed space $U$ into a normed space $V$. We have already learned that $\mathcal{L}(U, V)$ is a subspace of the space $L(U, V)$ of all linear (but not necessarily continuous) operators from $U$ to $V$, and that it can be equipped with the norm
$$|A|=|A|_{\mathcal{L}(U, V)}=\sup {\boldsymbol{u} \neq 0} \frac{|A \boldsymbol{u}|_V}{|\boldsymbol{u}|_U}$$ In the case of a finite-dimensional space $U$, the space $\mathcal{L}(U, V)$ simply coincides with $L(U, V)$ as every linear operator on $U$ is automatically continuous. In order to show this, consider an arbitrary basis $$\boldsymbol{e}_i, i=1,2, \ldots, n$$ for $U$ and a corresponding norm, $$|\boldsymbol{u}|=\sum{i=1}^n\left|u_i\right|, \text { where } \boldsymbol{u}=\sum_1^n u_i \boldsymbol{e}_i$$
As any two norms are equivalent in a finite-dimensional space (recall Exercise 4.6.3), it is sufficient to show that any linear operator on $U$ is continuous with respect to this particular norm. This follows easily from
\begin{aligned} |A \boldsymbol{u}|_V=\left|A\left(\sum_1^n u_i \boldsymbol{e}_i\right)\right| & \leq \sum_1^n\left|u_i\right|\left|A \boldsymbol{e}_i\right|_V \ & \leq\left(\max _i\left|A \boldsymbol{e}_i\right|_V\right) \sum_1^n\left|u_i\right| \end{aligned}

# 泛函分析代写

## 数学代写|泛函分析作业代写Functional Analysis代考|The Space of Continuous Linear Operators

$$|A|=|A|_{\mathcal{L}(U, V)}=\sup {\boldsymbol{u} \neq 0} \frac{|A \boldsymbol{u}|_V}{|\boldsymbol{u}|_U}$$在有限维空间$U$的情况下，空间$\mathcal{L}(U, V)$与$L(U, V)$重合，因为$U$上的每个线性算子都是自动连续的。为了说明这一点，考虑$U$的任意基$$\boldsymbol{e}_i, i=1,2, \ldots, n$$和相应的范数$$|\boldsymbol{u}|=\sum{i=1}^n\left|u_i\right|, \text { where } \boldsymbol{u}=\sum_1^n u_i \boldsymbol{e}_i$$

\begin{aligned} |A \boldsymbol{u}|_V=\left|A\left(\sum_1^n u_i \boldsymbol{e}_i\right)\right| & \leq \sum_1^n\left|u_i\right|\left|A \boldsymbol{e}_i\right|_V \ & \leq\left(\max _i\left|A \boldsymbol{e}_i\right|_V\right) \sum_1^n\left|u_i\right| \end{aligned}

## 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|Completeness and Completion of Metric Spaces

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。

## 数学代写|泛函分析作业代写Functional Analysis代考|Completeness and Completion of Metric Spaces

Cauchy Sequences. Every convergent sequence $x_n \in \mathbb{R}$ satisfies the so-called Cauchy condition
$$\forall \varepsilon>0 \quad \exists N:\left|x_n-x_m\right|<\varepsilon \quad \text { whenever } n, m \geq N$$ Indeed, let $x_0$ be the limit of $x_n$. Choose arbitrary $\epsilon>0$. There exists then $N$ such that, for every $n \geq N$, $\left|x_n-x_o\right|<\epsilon / 2$. Consequently, for $n, m \geq N$, $$\left|x_n-x_m\right|=\left|x_n-x_0+x_0-x_m\right| \leq\left|x_n-x_0\right|+\left|x_m-x_0\right|<\epsilon$$ Roughly speaking, when a sequence converges in $\mathbb{R}$, its entries $x_n, x_m$ get closer and closer together as $n, m$ increase. A sequence which satisfies the Cauchy condition is called a Cauchy sequence. Thus, every convergent sequence in $\mathbb{R}$ is a Cauchy sequence. It turns out that the converse is also true. THEOREM 4.8.1 Let $x_n \in \mathbb{R}$ be a Cauchy sequence. Then $x_n$ is convergent in $\mathbb{R}$, i.e., there exists $c \in \mathbb{R}$ such that $x_n \rightarrow c$. PROOF Consider the following two sets: \begin{aligned} & A:=\left{a \in \mathbb{R}: \exists N: n \geq N \Rightarrow a0 \quad \exists N \quad m, n \geq N \Rightarrow\left|x_n-x_m\right|<\epsilon

or, equivalently,
$$\forall \epsilon>0 \quad \exists N \quad m, n \geq N \Rightarrow x_m-\epsilon<x_n<x_m+\epsilon$$
which proves the following points about sets $A$ and $B$.

1. $\forall m \geq N x_m-\epsilon \in A, x_m+\epsilon \in B$. Consequently, sets $A, B$ are non-empty.
2. Elements of set $B$ provide upper bounds for set $A$ and, conversely, elements of $A$ provide lower bounds for set $B$.

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

Since in a metric space every point possesses a countable base of neighborhoods, according to Proposition 4.4.5, every compact set is sequentially compact. It turns out that, in the case of a metric space, the converse is also true.
THEOREM 4.9.1
(Bolzano-Weierstrass Theorem)
A set $E$ in a metric space $(X, d)$ is compact if and only if it is sequentially compact.
Before we prove this theorem, we shall introduce some auxiliary concepts.
$\varepsilon$-Nets and Totally Bounded Sets. Let $Y$ be a subset of a metric space $(X, d)$ and let $\varepsilon$ be a positive real number. A finite set
$$Y_{\varepsilon}=\left{y_{\varepsilon}^1, \ldots, y_{\varepsilon}^n\right} \subset X$$
is called an $\varepsilon$-net for $Y$ if
$$Y \subset \bigcup_{j=1}^n B\left(y_{\varepsilon}^j, \varepsilon\right)$$
In other words, for every $y \in Y$ there exists a point $y_{\varepsilon}^j \in Y_{\varepsilon}$ such that
$$d\left(y, y_{\varepsilon}^j\right)<\varepsilon$$ A set $Y \subset X$ is said to be totally bounded in $X$ if for each $\varepsilon>0$ there exists in $X$ an $\varepsilon$-net for $Y$. If $Y$ is totally bounded in itself, i.e., it contains the $\varepsilon$-nets, we say that $Y$ is totally bounded. Note that, in particular, every set $Y$ totally bounded in $X$ is bounded. Indeed, denoting by $M_{\varepsilon}$ the maximum distance between points in $\varepsilon$-net $Y_{\varepsilon}$
$$M_{\varepsilon}=\max \left{d(x, y): x, y \in Y_{\varepsilon}\right}$$
we have
$$d(x, y) \leq d\left(x, x^{\varepsilon}\right)+d\left(x^{\varepsilon}, y^{\varepsilon}\right)+d\left(y^{\varepsilon}, y\right) \leq M_{\varepsilon}+2 \varepsilon \quad \text { for every } x, y \in Y$$
where $x^{\varepsilon}$ and $y^{\varepsilon}$ are points from $\varepsilon$-net $Y_{\varepsilon}$ such that
$$d\left(x, x^{\varepsilon}\right)<\varepsilon \text { and } d\left(y, y^{\varepsilon}\right)<\varepsilon$$
Consequently, $\operatorname{dia} Y \leq M_{\varepsilon}+2 \varepsilon$, which proves that $Y$ is bounded.

# 泛函分析代写

## 数学代写|泛函分析作业代写Functional Analysis代考|Completeness and Completion of Metric Spaces

$$\forall \varepsilon>0 \quad \exists N:\left|x_n-x_m\right|<\varepsilon \quad \text { whenever } n, m \geq N$$的确，让$x_0$成为$x_n$的极限。任意选择$\epsilon>0$。因此存在$N$，对于每一个$n \geq N$, $\left|x_n-x_o\right|<\epsilon / 2$。因此，对于$n, m \geq N$, $$\left|x_n-x_m\right|=\left|x_n-x_0+x_0-x_m\right| \leq\left|x_n-x_0\right|+\left|x_m-x_0\right|<\epsilon$$粗略地说，当一个序列收敛于$\mathbb{R}$时，随着$n, m$的增加，它的条目$x_n, x_m$会越来越接近。满足柯西条件的序列称为柯西序列。因此，$\mathbb{R}$中的每一个收敛序列都是柯西序列。反过来也是对的。定理4.8.1设$x_n \in \mathbb{R}$为柯西序列。那么$x_n$收敛于$\mathbb{R}$，即存在$c \in \mathbb{R}$使得$x_n \rightarrow c$。请考虑以下两组:\begin{aligned} & A:=\left{a \in \mathbb{R}: \exists N: n \geq N \Rightarrow a0 \quad \exists N \quad m, n \geq N \Rightarrow\left|x_n-x_m\right|<\epsilon

$$\forall \epsilon>0 \quad \exists N \quad m, n \geq N \Rightarrow x_m-\epsilon<x_n<x_m+\epsilon$$

$\forall m \geq N x_m-\epsilon \in A, x_m+\epsilon \in B$． 因此，集合$A, B$是非空的。

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

(Bolzano-Weierstrass定理)

$\varepsilon$-网和完全有界集合。让 $Y$ 是度量空间的子集 $(X, d)$ 让 $\varepsilon$ 是一个正实数。有限集
$$Y_{\varepsilon}=\left{y_{\varepsilon}^1, \ldots, y_{\varepsilon}^n\right} \subset X$$

$$Y \subset \bigcup_{j=1}^n B\left(y_{\varepsilon}^j, \varepsilon\right)$$

$$d\left(y, y_{\varepsilon}^j\right)<\varepsilon$$ 一套 $Y \subset X$ 是完全被限定的吗 $X$ 如果是每个 $\varepsilon>0$ 存在于 $X$ 一个 $\varepsilon$-net for $Y$． 如果 $Y$ 是完全局限于自身的，也就是说，它包含了 $\varepsilon$-nets，我们这么说 $Y$ 是完全有界的。特别要注意的是，每个集合 $Y$ 完全局限于 $X$ 是有界的。的确，用 $M_{\varepsilon}$ 点之间的最大距离 $\varepsilon$-net $Y_{\varepsilon}$

$$M_{\varepsilon}=\max \left{d(x, y): x, y \in Y_{\varepsilon}\right}$$

$$d(x, y) \leq d\left(x, x^{\varepsilon}\right)+d\left(x^{\varepsilon}, y^{\varepsilon}\right)+d\left(y^{\varepsilon}, y\right) \leq M_{\varepsilon}+2 \varepsilon \quad \text { for every } x, y \in Y$$

$$d\left(x, x^{\varepsilon}\right)<\varepsilon \text { and } d\left(y, y^{\varepsilon}\right)<\varepsilon$$

## 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|拓扑学代写Topology代考|Quotients by Properly Discontinuous Actions

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

## 数学代写|拓扑学代写Topology代考|Quotients by Properly Discontinuous Actions

Definition 12.15 Let $G$ be a subgroup of the group $\operatorname{Homeo}(E)$ of homeomorphisms of a space $E$. The group $G$ is said to act properly discontinuously if every point $e \in E$ has a neighbourhood $U$ such that $g(U) \cap U=\emptyset$ for any $g \in G$ different from the identity.

isomorphic to $\mathbb{Z}$, that acts properly discontinuously.
Example 12.17 The subgroup in Homeo $\left(\mathbb{R}^2-{0}\right)$ generated by the multiplication by a number $\lambda>1$ acts in a properly discontinuous fashion. acting properly discontinuously. If $E / G$ is connected, then the quotient map $p: E \rightarrow$ $E / G$ is a covering map.

Proof Fix $e \in E$ and choose an open set $U \subset E$ such that $e \in U$ and $g(U) \cap U=\emptyset$ for every $g$ different from the identity.
Proposition 5.15 implies that $p: E \rightarrow E / G$ is an open map, and
$$p^{-1}(p(U))=\cup{g(U) \mid g \in G}$$
So we just need to prove that, for any $g \in G$, the open sets $g(U)$ are disjoint and that $p: g(U) \rightarrow p(U)$ is a homeomorphism.

Since $g(U) \cap h(U)=h\left(h^{-1} g(U) \cap U\right)$, it follows $g(U) \cap h(U)=\emptyset$ for every $g \neq$ $h$. The quotient map $p: U \rightarrow p(U)$ is open and bijective hence a homeomorphism. The map $p: g(U) \rightarrow p(U)$ is the composite of the homeomorphisms $g^{-1}: g(U) \rightarrow$ $U$ with $p: U \rightarrow p(U)$.

## 数学代写|拓扑学代写Topology代考|Lifting Homotopies

Definition 12.23 Let $f: Y \rightarrow X$ be a continuous map and $p: E \rightarrow X$ a covering space. A continuous mapping $g: Y \rightarrow E$ is called a lift of $f$ when the diagram commutes, i.e. $f=p g$.
Lemma 12.24 For any covering space $p: E \rightarrow X$ the diagonal $\Delta \subset E \times E$ is open and closed in the fibred product
$$E \times_X E={(u, v) \in E \times E \mid p(u)=p(v)} .$$
Proof Take $(e, e) \in \Delta$ and choose an open set $U \subset E$ such that $e \in U$ and the restriction $p: U \rightarrow X$ is $1-1$. Then
$$(U \times U) \cap\left(E \times_X E\right)=U \times_X U$$
is an open neighbourhood of $(e, e)$ in the fibred product. On the other hand
$$(U \times U) \cap\left(E \times_X E\right)={(u, v) \in U \times U \mid p(u)=p(v)} \subset \Delta,$$
proving that $\Delta$ is a neighbourhood of any of its points, inside the fibred product.
Conversely, if $\left(e_1, e_2\right) \in E \times_X E-\Delta$ we pick an admissible open set $V$ containing $p\left(e_1\right)=p\left(e_2\right)$. Since $e_1 \neq e_2$, there exist disjoint open sets $U_1, U_2 \subset p^{-1}(V)$ such that $e_1 \in U_1, e_2 \in U_2$. Therefore
$$\left(e_1, e_2\right) \in\left(U_1 \times U_2\right) \cap\left(E \times_X E\right) \subset E \times_X E-\Delta,$$
so that the diagonal is closed in the fibred product.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Quotients by Properly Discontinuous Actions

$$\pi_1(f): \pi_1(X, a) \rightarrow \pi_1(Y, f(a)), \quad \pi_1(f)([\alpha])=[f \alpha]$$

$$F: I^2 \rightarrow X, \quad F(t, s)=R(\beta(t), s)$$

## 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|拓扑学代写Topology代考|Locally Connected Spaces and the Functor π0

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

## 数学代写|拓扑学代写Topology代考|Locally Connected Spaces and the Functor π0

Definition 10.1 A space is locally connected if every point has local basis of connected neighbourhoods.

From Lemma 4.28 the connected components of a locally connected space are open. While general connected spaces may not be locally connected (Exercise 10.1), open sets in $\mathbb{R}^n$ are locally connected, and the product of two locally connected spaces is locally connected.

Definition 10.2 Let $X$ be a topological space. Denote by $\pi_0(X)=X / \sim$ the quotient space under the relation $\sim$ that identifies points connected by a path in $X$.

To be more precise, for any two points $x, y \in X$ one defines the set of paths from $x$ to $y$ :
$$\Omega(X, x, y)={\alpha:[0,1] \rightarrow X \mid \alpha \text { continuous, } \alpha(0)=x, \alpha(1)=y}$$
and then
$$\pi_0(X)=X / \sim, \quad \text { where } \quad x \sim y \Longleftrightarrow \Omega(X, x, y) \neq \emptyset$$
We have to make sure $\sim$ is an equivalence relation.
Reflexivity. To prove $x \sim x$ we consider the constant path
$$1_x:[0,1] \rightarrow X, \quad 1_x(t)=x \text { for every } t \in[0,1] .$$
Symmetry. For every $x, y \in X$ we have the path-reverting operator
$$i: \Omega(X, x, y) \rightarrow \Omega(X, y, x), \quad i(\alpha)(t)=\alpha(1-t),$$
that is clearly invertible. In particular $\Omega(X, x, y)$ is empty precisely when $\Omega(X, y, x)$ is empty.
Transitivity. We just consider the product of paths (or composite)
$$*: \Omega(X, x, y) \times \Omega(X, y, z) \rightarrow \Omega(X, x, z), \quad(\alpha, \beta) \mapsto \alpha * \beta,$$
where
$$\alpha * \beta(t)= \begin{cases}\alpha(2 t) & \text { if } 0 \leq t \leq 1 / 2 \ \beta(2 t-1) & \text { if } 1 / 2 \leq t \leq 1\end{cases}$$

## 数学代写|拓扑学代写Topology代考|Homotopy

Definition 10.8 Two continuous maps $f_0, f_1: X \rightarrow Y$ are said to be homotopic if there is a continuous function
$$F: X \times[0,1] \rightarrow Y$$
such that $F(x, 0)=f_0(x)$ and $F(x, 1)=f_1(x)$ for every $x \in X$. Such an $F$ is called a homotopy between $f_0$ and $f_1$.

To help one ‘visualise’ the meaning of the above definition let’s write $f_t(x)=$ $F(x, t)$ for every $(x, t) \in X \times[0,1]$. Then for any $t \in[0,1]$ the map
$$f_t: X \rightarrow Y$$
is continuous. When $t=0$ we recover $f_0$, which deforms in a continuous way, as $t$ varies, until it becomes $f_1$ for $t=1$.

Example 10.9 Let $Y \subset \mathbb{R}^n$ be a convex subspace. For any topological space $X$, two continuous maps $f_0, f_1: X \rightarrow Y$ are homotopic: it suffices to define the homotopy as
$$F: X \times[0,1] \rightarrow Y, \quad F(x, t)=(1-t) f_0(x)+t f_1(x)$$

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Locally Connected Spaces and the Functor π0

$$\Omega(X, x, y)={\alpha:[0,1] \rightarrow X \mid \alpha \text { continuous, } \alpha(0)=x, \alpha(1)=y}$$

$$\pi_0(X)=X / \sim, \quad \text { where } \quad x \sim y \Longleftrightarrow \Omega(X, x, y) \neq \emptyset$$

$$1_x:[0,1] \rightarrow X, \quad 1_x(t)=x \text { for every } t \in[0,1] .$$

$$i: \Omega(X, x, y) \rightarrow \Omega(X, y, x), \quad i(\alpha)(t)=\alpha(1-t),$$

$$*: \Omega(X, x, y) \times \Omega(X, y, z) \rightarrow \Omega(X, x, z), \quad(\alpha, \beta) \mapsto \alpha * \beta,$$

$$\alpha * \beta(t)= \begin{cases}\alpha(2 t) & \text { if } 0 \leq t \leq 1 / 2 \ \beta(2 t-1) & \text { if } 1 / 2 \leq t \leq 1\end{cases}$$

## 数学代写|拓扑学代写Topology代考|Homotopy

$$F: X \times[0,1] \rightarrow Y$$

## 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。