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## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Definitions and first examples

Let $X$ and $Y$ be two sets. A function $f$ defined on $X$ with values in $Y$ is a correspondence associating to each element $x \in X$ at most one element $y \in Y$. This is often shortened to ‘a function from $X$ to $Y$ ‘. A synonym for function is map. The set of $x \in X$ to which $f$ associates an element in $Y$ is the domain of $f$; the domain is a subset of $X$, indicated by $\operatorname{dom} f$. One writes
$$f: \operatorname{dom} f \subseteq X \rightarrow Y$$
If $\operatorname{dom} f=X$, one says that $f$ is defined on $X$ and writes simply $f: X \rightarrow Y$.
The element $y \in Y$ associated to an element $x \in \operatorname{dom} f$ is called the image of $x$ by or under $f$ and denoted $y=f(x)$. Sometimes one writes
$$f: x \mapsto f(x)$$
The set of images $y=f(x)$ of all points in the domain constitutes the range of $f$, a subset of $Y$ indicated by $\operatorname{im} f$.

The graph of $f$ is the subset $\Gamma(f)$ of the Cartesian product $X \times Y$ made of pairs $(x, f(x))$ when $x$ varies in the domain of $f$, i.e.,
$$\Gamma(f)={(x, f(x)) \in X \times Y: x \in \operatorname{dom} f}$$

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Range and pre-image

Let $A$ be a subset of $X$. The image of $A$ under $f$ is the set
$$f(A)={f(x): x \in A} \subseteq \operatorname{im} f$$
of all the images of elements of $A$. Notice that $f(A)$ is empty if and only if $A$ contains no elements of the domain of $f$. The image $f(X)$ of the whole set $X$ is the range of $f$, already denoted by $\operatorname{im} f$.
Let $y$ be any element of $Y$; the pre-image of $y$ by $f$ is the set
$$f^{-1}(y)={x \in \operatorname{dom} f: f(x)=y}$$
of elements in $X$ whose image is $y$. This set is empty precisely when $y$ does not belong to the range of $f$. If $B$ is a subset of $Y$, the pre-image of $B$ under $f$ is defined as the set
$$f^{-1}(B)={x \in \operatorname{dom} f: f(x) \in B}$$
union of all pre-images of elements of $B$.
It is easy to check that $A \subseteq f^{-1}(f(A))$ for any subset $A$ of $\operatorname{dom} f$, and $f\left(f^{-1}(B)\right)=B \cap \operatorname{im} f \subseteq B$ for any subset $B$ of $Y$.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Definitions and first examples

$$f: \operatorname{dom} f \subseteq X \rightarrow Y$$

$$f: x \mapsto f(x)$$

$$\Gamma(f)={(x, f(x)) \in X \times Y: x \in \operatorname{dom} f}$$

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Range and pre-image

$$f(A)={f(x): x \in A} \subseteq \operatorname{im} f$$

$$f^{-1}(y)={x \in \operatorname{dom} f: f(x)=y}$$

$$f^{-1}(B)={x \in \operatorname{dom} f: f(x) \in B}$$
$B$所有元素的预映像的并集。

## 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。

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## avatest™帮您通过考试

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## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|The ordering of real numbers

Non-zero real numbers are either positive or negative. Positive reals form the subset $\mathbb{R}{+}$, negative reals the subset $\mathbb{R}{-}$. We are thus in presence of a partition $\mathbb{R}=\mathbb{R}{-} \cup{0} \cup \mathbb{R}{+}$. The set
$$\mathbb{R}={0} \cup \mathbb{R}{+}$$
of non-negative reals will also be needed. Positive numbers correspond to points on the line lying at the right – with respect to the positive direction – of the origin.
Instead of $x \in \mathbb{R}{+}$, one simply writes $x>0$ (‘ $x$ is bigger, or larger, than 0 ‘); similarly, $x \in \mathbb{R}$ will be expressed by $x \geq 0$ (‘ $x$ is bigger or equal than 0 ‘). Therefore an order relation is defined by
$$x0 .$$
This is a total ordering, i.e., given any two distinct reals $x$ and $y$, one (and only one) of the following holds: either $x<y$ or $y<x$. From the geometrical point of view the relation $x<y$ tells that the point with coordinate $x$ is placed at the left of the point with coordinate $y$. Let us also define
$$x \leq y \quad \Longleftrightarrow \quad x<y \quad \text { or } \quad x=y$$

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Absolute value

Let us introduce now a simple yet important notion. Given a real number $x$, one calls absolute value of $x$ the real number
$$|x|= \begin{cases}x & \text { if } x \geq 0 \ -x & \text { if } x<0\end{cases}$$ Thus $|x| \geq 0$ for any $x$ in $\mathbb{R}$. For instance $|5|=5,|0|=0,|-5|=5$. Geometrically, $|x|$ represents the distance from the origin of the point with coordinate $x$; thus, $|x-y|=|y-x|$ is the distance between the two points of coordinates $x$ and $y$. The following relations, easy to prove, will be useful $$|x+y| \leq|x|+|y|, \quad \text { for all } x, y \in \mathbb{R}$$ (called triangle inequality) and $$|x y|=|x||y|, \quad \text { for all } x, y \in \mathbb{R} \text {. }$$ Throughout the text we shall solve equations and inequalities involving absolute values. Let us see the simplest ones. According to the definition, $$|x|=0$$ has the unique solution $x=0$. If $a$ is any number $>0$, the equation
$$|x|=a$$
has two solutions $x=a$ and $x=-a$, so
$$|x|=a \quad \Longleftrightarrow \quad x= \pm a, \quad \forall a \geq 0 .$$

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|The ordering of real numbers

$$\mathbb{R}={0} \cup \mathbb{R}{+}$$

$$x0 .$$

$$x \leq y \quad \Longleftrightarrow \quad x<y \quad \text { or } \quad x=y$$

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Absolute value

$$|x|= \begin{cases}x & \text { if } x \geq 0 \ -x & \text { if } x<0\end{cases}$$因此对于$\mathbb{R}$中的任何$x$都是$|x| \geq 0$。例如$|5|=5,|0|=0,|-5|=5$。几何上，$|x|$表示点到原点的距离，坐标为$x$;因此，$|x-y|=|y-x|$为坐标$x$和$y$的两点之间的距离。下面的关系，很容易证明，将是有用的$$|x+y| \leq|x|+|y|, \quad \text { for all } x, y \in \mathbb{R}$$(称为三角不等式)和$$|x y|=|x||y|, \quad \text { for all } x, y \in \mathbb{R} \text {. }$$在整个文本中，我们将解决方程和不等式涉及绝对值。让我们看看最简单的。根据定义，$$|x|=0$$有唯一解$x=0$。如果$a$是任意数$>0$，方程
$$|x|=a$$

$$|x|=a \quad \Longleftrightarrow \quad x= \pm a, \quad \forall a \geq 0 .$$

## 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。

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## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Fubini’s Theorem

Theorem 8.8.10 (Tonelli’s theorem). Suppose $f: X \times Y \rightarrow \mathbb{C}$ is an $\mathfrak{M} \otimes \mathfrak{R}$ measurable function.
(a) If $f$ is positive, let $\varphi(x)=\int_Y f_x d v$, and $\psi(y)=\int_X f^{\prime} d \mu$. Then $\varphi$ is $\mathfrak{M}$ measurable, $\psi$ is $\Re$-measurable, and
$$\int_X \varphi d \mu=\int_{X \times Y} f d(\mu \otimes \nu)=\int_Y \psi d \nu .$$
(b) In general, let $\varphi^(x)=\int_Y|f|_x d \nu$, and $\psi^(y)=\int_X|f|^y d \mu$. If $\varphi^* \in \mathbb{Q}^1(\mu)$ or if $\psi^* \in \mathbf{\Omega}^1(\nu)$, then $f \in \mathbf{\Omega}^1(\mu \otimes \nu)$.

Proof. Tonelli’s theorem holds for the characteristic function of an $\mathfrak{M} \otimes \mathfrak{N}$ measurable set by the previous theorem. By the linearity of the integral, Tonelli’s theorem holds for any $\mathfrak{M} \otimes \mathfrak{N}$-measurable simple function.

Now let $0 \leq s_1 \leq s_2 \leq \ldots$ be a sequence of $\mathfrak{M} \otimes \mathfrak{N}$-simple functions converging to $f(x, y)$ for every $(x, y) \in X \times Y$, and let $\varphi_n(x)=\int_Y\left(s_n\right)x d \nu$. By the above paragraph, $\int_X \varphi_n d \mu=\int{X \times Y} s_n d(\mu \otimes \nu)$. The monotone convergence theorem implies that $\int_X \varphi d \mu=\int_{X \times Y} f d(\mu \otimes \nu)$. The proof that $\int_Y \psi d \nu=\int_{X \times Y} f d(\mu \otimes \nu)$ is identical to the above.
Part (b) is obtained by applying part (a) to the function $|f|$.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Products of Lebesgue Measures

In the discussion below and until the end of the section, $k$ is a positive integer, and $\lambda_k$ denotes Lebesgue measure on the $\sigma$-algebra $\mathcal{L}^k$ of Lebesgue measurable subsets of $\mathbb{R}^k$. We also use the notation $\mathcal{B}^k$ to denote the $\sigma$-algebra of Borel subsets of $\mathbb{R}^k$.
In the following, we use the result of problem 10 on section 8.4 without explicit mention.

Lemma 8.8.12. Let $r$ and $s$ be positive integers, and let $n=r+s$. If $Z$ is a set of Lebesgue measures 0 in $\mathbb{R}^r$ and $B \in \mathcal{L}^s$, then $Z \times B \in \mathcal{L}^n$, and $\lambda_n(Z \times B)=0$.
Proof. First assume that $B$ is bounded, and choose an open set $V$ of finite measure such that $B \subseteq V \subseteq \mathbb{R}^s$. Let $\in>0$. Choose an open set $U$ such that $Z \subseteq U \subseteq \mathbb{R}^r$ and $\lambda_r(U)<\epsilon$. Since we have not yet established the Lebesgue measurability of $Z \times B$, we estimate its outer measure: $m_n^(Z \times B) \leq m_n^(U \times V)=\lambda_n(U \times V)=$ $\lambda_r(U) \lambda_s(V)<\epsilon \lambda_s(V)$. Since $\epsilon$ is arbitrary, $m_n^*(Z \times B)=0$; hence $Z \times B$ is measurable of measure 0 .

If $B$ is unbounded, consider the intersection $B_i$ of $B$ with the open ball in $\mathbb{R}^s$ of radius $i$ and centered at the origin. By what we just proved, for each $i \in \mathbb{N}$, $Z \times B_i \in \mathcal{L}^n$ has measure 0 . Since $Z \times B=\cup_{i=1}^{\infty}\left(Z \times B_i\right)$, the proof is complete.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Fubini’s Theorem

(a)如果$f$是正的，设$\varphi(x)=\int_Y f_x d v$，和$\psi(y)=\int_X f^{\prime} d \mu$。那么$\varphi$是$\mathfrak{M}$可测量的，$\psi$是$\Re$可测量的，和
$$\int_X \varphi d \mu=\int_{X \times Y} f d(\mu \otimes \nu)=\int_Y \psi d \nu .$$
(b)一般来说，让$\varphi^(x)=\int_Y|f|_x d \nu$和$\psi^(y)=\int_X|f|^y d \mu$。如果$\varphi^* \in \mathbb{Q}^1(\mu)$或者$\psi^* \in \mathbf{\Omega}^1(\nu)$，那么$f \in \mathbf{\Omega}^1(\mu \otimes \nu)$。

## 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:Mathematical Analysis, 数学代写, 数学分析

## 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|数学分析代写MATHEMATICAL ANALYSIS代考|Orthonormal Bases and Fourier Series

In the introduction to section 7.1, we made the case for the existence of a maximal orthonormal sequence $\left{u_1, u_2, \ldots\right}$ in a Hilbert space $H$. As you will see in this section, some Hilbert spaces do not admit countable maximal orthonormal subsets. Perhaps we must first tackle the problem of the existence of a maximal orthogonal subset of $H$, then examine the problem of which Hilbert spaces possess a countable such subset. In this section, we provide solutions to both problems and reveal the basic structure of a Hilbert space, hence paving the way to answer the problems posed in section 4.10 .
The proof of the following theorem can be seen in section 3.7.
Theorem 7.2.1. An orthogonal subset $S$ of a Hilbert space $H$ is independent.
Definition. An orthonormal basis for a Hilbert space $H$ is a maximal orthonormal subset of $H$. An orthonormal subset of $H$ is maximal if it is not properly contained in another orthonormal subset of $H$.
Example 1. We show that $\left{e_n: n \in \mathbb{N}\right}$ is an orthonormal basis for $l$. It is clear that $S$ is orthonormal. If $x=\left(x_n\right) \in l^2$ is orthogonal to $S$, then, for every $n \in \mathbb{N}, x_n=\left\langle x, e_n\right\rangle=0$, and hence $x=0$.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Excursion: Inseparable Hilbert Spaces

Inseparable Hilbert spaces do exist. They are mostly a curiosity and do not have much practical use. We include the discussion below for the satisfaction of the inquisitive reader.

The motivation for the definition below and the construction in theorem 7.2.9 is provided by the following example.

Example 6. Let $S=\left{u_\alpha: \alpha \in I\right}$ be an uncountable orthonormal subset of a Hilbert space $H$. For a vector $x \in H$, consider the set of Fourier coefficients $\left{\hat{x}\alpha: \alpha \in I\right}$. We claim that $\hat{x}\alpha=0$ for all but countably many $\alpha \in I$.

Let $\left{u_{\alpha_1}, \ldots, u_{\alpha_n}\right}$ be a finite subset of $S$. By theorem 7.2.5, $\sum_{i=1}^n\left|\hat{x}{\alpha_i}\right|^2 \leq$ $|x|^2<\infty$. It follows that $\sum{\alpha \in I}\left|\hat{x}\alpha\right|^2<\infty$ (see example 1 in section 4.10 and the definition preceding it); hence the set $\left{\alpha \in I: \hat{x}\alpha \neq 0\right}$ is countable.
The above example strongly suggests the following definition.
Definition. Let $I$ be an infinite set, and let $\aleph=\operatorname{Card}(I)$. Define $l^2(\aleph)$ to be the set of all functions $x: I \rightarrow \mathbb{C}$ such that $x_\alpha=0$ for all but countably many $\alpha \in I$ and $|x|=\left(\sum_{\alpha \in I}\left|x_\alpha\right|^2\right)^{1 / 2}<\infty$. To eliminate any danger of ambiguity, let $I_x=$ $\left{\alpha_1, \alpha_2, \ldots\right}$ be the subset of $I$ for which $x_\alpha \neq 0$. The notation $\sum_{\alpha \in I}\left|x_\alpha\right|^2$ means $\sum_{i=1}^{\infty}\left|x_{\alpha_i}\right|^2$. We will continue to employ this notation for the remainder of this discussion.

Theorem 7.2.9. The set $\mathcal{R}(\aleph)$ is a Hilbert space with the operations defined within the proof.

Proof. Let $x=\left(x_\alpha\right)$, and $y=\left(y_\alpha\right) \in l^2(\aleph)$. We show that $x+y \in l^2(\aleph)$ and that $|x+y| \leq|x|+|y|$. Let $I_x=\left{\alpha \in I: x_\alpha \neq 0\right}$, and $I_y=\left{\alpha \in I: y_\alpha \neq 0\right}$, and let $J=I_x \cup I_y$. Since $J$ is countable, we can write $J=\left{\alpha_1, \alpha_2, \ldots\right}$. Note that $\hat{x}=\left(x_{\alpha_i}\right)$ and $\hat{y}=\left(y_{\alpha_i}\right)$ are in $l^2$; hence $|\hat{x}+\hat{y}|_2 \leq|\hat{x}|_2+|\hat{y}|_2$. But every $\alpha$ for which $x_\alpha+y_\alpha \neq 0$ is in J; hence $|x+y|=\left(\sum_{i=1}^{\infty}\left|x_{\alpha_i}+y_{\alpha_i}\right|^2\right)^{1 / 2}=|\hat{x}+\hat{y}|_2 \leq|\hat{x}|_2+$ $|\hat{y}|_2=|x|+|y|$. The fact that $|a x|=|a||x|$ for all $x \in \mathcal{l}^2(\aleph)$ and all scalars $a$ requires an even simpler argument. The rest of the properties of a normed linear space are easily verifiable. Thus $\mathcal{L}^{\mathcal{}}(\mathcal{N})$ is a normed linear space.

Define an inner product on $l^2(\aleph)$ as follows: $\langle x, y\rangle=\langle\hat{x}, \hat{y}\rangle=\sum_{i=1}^{\infty} x_{\alpha_i} \bar{y}{\alpha_i}$ This inner product induces the norm on $\mathcal{L}^2(\aleph)$ we defined earlier. We now show the completeness of $l^2(\aleph)$. Suppose $\left(x^{(n)}\right)$ is a Cauchy sequence in $l^2(\aleph)$, let $I_n=\left{\alpha \in I: x\alpha^{(n)} \neq 0\right}$, and let $J=\cup_{n \in \mathbb{N}} I_n$. Then $J$ is a countable subset of $I$, and we can write $J=\left{\alpha_1, \alpha_2, \ldots\right}$. Since $\left|\hat{x}^{(m)}-\hat{y}^{(n)}\right|=\left|x^{(n)}-y^{(n)}\right|,\left(\hat{x}^{(n)}\right)$ is a Cauchy sequence in $l^2$ and is therefore convergent to an element $\hat{x}=\left(x_1, x_2, \ldots\right) \in R^2$. Define $x \in R^2(\aleph)$ by
$$x_\alpha= \begin{cases}x_i & \text { if } \alpha=\alpha_i, \ 0 & \text { otherwise. }\end{cases}$$
Clearly, $x^{(n)}$ converges to $x$ in $\mathcal{R}^2(\boldsymbol{\aleph})$.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Excursion: Inseparable Hilbert Spaces

$$x_\alpha= \begin{cases}x_i & \text { if } \alpha=\alpha_i, \ 0 & \text { otherwise. }\end{cases}$$

## 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。

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## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Three Fundamental Theorems

In addition to the Hahn-Banach theorem, the three theorems we present in this section are of fundamental importance. All three theorems require completeness; hence they apply only to Banach spaces.
In chapter 4 (see problem 5 on section 4.8 ), we encountered an example where a family of pointwise bounded functions on a complete metric space is, in fact, uniformly bounded on a ball. Lemma 6.3.1 is similar in spirit, and its proof demonstrates the centrality of Baire’s theorem in this section. Because the boundedness of a linear function on a ball implies its boundedness, it must not be surprising that when $X$ is a Banach space, pointwise boundedness implies uniform boundedness. This is the uniform boundedness principle.
The open mapping theorem is a central theorem in functional analysis, and one cannot exaggerate its importance. Lemma 6.3 .3 is critical to the proof of the open mapping theorem, and, again, completeness is crucial. The closed graph theorem comes in quite handy in certain applications to prove the boundedness of a linear function. It follows rather easily from the open mapping theorem. Later in the book, you will see many applications of the three theorems, as well as the Hahn-Banach theorem.
In this section, $X$ and $Y$ are normed linear spaces.
A family of bounded linear functions $\left{T_\alpha\right}_{\alpha \in I}$ from $X$ to $Y$ such that, for each $x \in X$, $\sup {\alpha \in I}\left{\left|T\alpha(x)\right|\right}<\infty$ is said to be pointwise bounded. If $\sup {\alpha \in I}\left|T\alpha\right|<\infty$, we say that the family $\left{T_\alpha\right}$ is uniformly bounded.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|The Hahn-Banach Theorem

The importance of the Hahn-Banach theorem cannot be overstated. The results following theorem 6.4.4 represent only a sample of the wide range of applications of the Hahn-Banach theorem. Unlike the three major theorems of the previous section, the Hahn-Banach theorem does not require completeness.
The Hahn-Banach theorem has many guises, and one of them is an extension theorem. The following example shows that, from the purely algebraic perspective, extending a linear functional on a subspace $M$ of a vector space $X$ is a trivial task. Compare the following example to theorem 6.4.4.
Example 1. Let $M$ be a subspace of a vector $\operatorname{space} X$, and let $\lambda$ be a linear functional on $M$. Then $\lambda$ can be extended to a linear functional on $X$.
Let $S_1$ be a basis for $M$, and choose a subset $S_2$ of $X$ such that $S_1 \cup S_2$ is a basis for $X$. Define a function $\Lambda: S \rightarrow \mathbb{C}$ as follows:
$$\Lambda(x)= \begin{cases}\lambda(x) & \text { if } x \in S_1, \ 0 & \text { if } x \in S_2\end{cases}$$
Extend the function $\Lambda$ by linearity to a functional $\Lambda$ on $X$. The restriction of $\Lambda$ to $M$ is clearly $\lambda$.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|The Hahn-Banach Theorem

$$\Lambda(x)= \begin{cases}\lambda(x) & \text { if } x \in S_1, \ 0 & \text { if } x \in S_2\end{cases}$$

## 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。

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## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Second Countable Spaces

In this section, we study second countable, separable, and Lindelöf spaces. Theorem 4.5.1 states that all three conditions are equivalent for metric spaces. This is not true for general topological spaces, and several counterexamples are provided in this section and the section exercises to show the nonequivalence of the three conditions. However, second countability implies the other two conditions. Second countability has other pleasant consequences, especially when it is combined with normality or local compactness. The definitions in this section are identical to the those in the metric case and are included below for ease of reference.
Definition. A subset $A$ of a topological space $X$ is dense in $X$ if $\bar{A}=X$.
Definition. A topological space $X$ is separable if it contains a countable dense subset.

Definition. A topological space $X$ is second countable if the topology on $X$ contains a countable open base.

Definition. A topological space $X$ is said to be a Lindelöf space if every open cover of $X$ contains a countable subcover of $X$. The definitions of open covers and subcovers can be seen in section 4.5.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Compact Spaces

In section 4.7 , we studied compact metric spaces extensively, including several equivalent formulations of the definition of compactness. We adopt the same definition of compactness in this chapter because the other characterizations do not lend themselves easily to generalization to general topological spaces, and especially because some of the other characterizations of compact metric spaces are false in general. You will also see that compact spaces have pleasant separation properties. Finally, we will prove the celebrated Tychonoff theorem for the product of finitely many topological spaces. The leading theorems in this section have counterparts in section 4.7. Therefore, proofs that duplicate those in section 4.7 will be omitted.

Definition. A topological space $X$ is said to be compact if every open cover of $X$ contains a finite subcover of $X$.
Example 1. The co-finite topology on an infinite set $X$ is compact.
Let $\mathcal{U}$ be an open cover of $X$, and fix an element $U_1 \in \mathcal{U}$. The complement of $U_1$ is finite, say, $U_1=X-\left{x_2, \ldots, x_n\right}$. Now, for each $2 \leq i \leq n$, pick an element $U_i \in \mathcal{U}$ that contains $x_i$. The finite collection $\left{U_1, \ldots, U_n\right}$ covers $X$.

Example 2. A real-valued, locally bounded function $f$ on a compact space $X$ is bounded.

By the definition of local boundedness, for every $x \in X$, there exists a positive number $M_x$ and an open neighborhood $U_x$ of $x$ such that $\sup {x \in U_x}|f(x)| \leq M_x$. Clearly, $\left{U_x: x \in X\right}$ is an open cover of $X$. Choose points $x_1, \ldots, x_n \in X$ such that the sets $U{x_1}, \ldots, U_{x_n}$ cover $X$, and let $M=\max {1 \leq i \leq n} M{x_i}$. For $x \in X, x \in U_{x_i}$ for some $1 \leq i \leq n$, and $|f(x)| \leq M_{x_i} \leq M$.

## 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。

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## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Definitions and Basic Properties

While the metric topology is often sufficient for most introductory courses in analysis, a good understanding of the elements of general topology is essential for any advanced study of analysis. An attempt to define topology in a paragraph is quite difficult and not likely to be successful, but we offer the following narrative for the satisfaction of the the reader who insists on an overview of the subject. We saw in chapter 4 that the collection of open sets generated by a metric has many intrinsic properties independent of the defining metric. In this section, we study the arrangement of the collection of open sets, or the topology, in a metric-free context. Every metric space is a topological space; hence all results for topological spaces (which are meaningful in the metric setting) are also valid for metric spaces, but not conversely. We often fall back on the metric case to gain insight into both subjects. We will encounter in this section many of the definitions that appeared in chapter 4, such as closure, interior, and boundary. We include those definitions again in this chapter for ease of reference. However, the proofs that duplicate those in chapter 4 are omitted. The amount of duplication is small and does not rise to the level of redundancy. We encourage the reader to compare results in this section to their counterparts in the previous chapter. The exercise is insightful.
Let $X$ be a nonempty set, and let $\mathcal{T}$ be a collection of subsets of $X ; \mathcal{T}$ is called a topology on $X$ if
(a) $\varnothing$ and $X$ are in $\mathcal{T}$,
(b) the union of an arbitrary family of members of $\mathcal{T}$ is a member of $\mathcal{T}$, and
(c) the intersection of two members of $\mathcal{T}$ is a member of $\mathcal{T}$.
Thus $\mathcal{T}$ is closed under the formation of arbitrary unions and finite intersections. The members of $\mathcal{T}$ are called the open subsets of $X$, and the pair $(X, \mathcal{T})$ is called a topological space.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Subspace Topology

Let $(X, \mathcal{J})$ be a topological space, and let $Y$ be a subset of $X$. Define $\mathcal{T}Y={Y \cap U$ : $U \in \mathcal{T}}$. It is easy to verify that $\mathcal{J}_Y$ is a topology on $Y$. For example, if $\left{Y \cap U\alpha\right}_\alpha$ is a collection of members of $\mathcal{T}Y$, then $\cup\alpha\left(Y \cap U_\alpha\right)=Y \cap\left(\cup_\alpha U_\alpha\right)$, which is in $\mathcal{T}Y$ because $\cup\alpha U_\alpha \in \mathcal{T}$. Verifying that $\mathcal{J}_Y$ is closed under the formation of finite intersections is straightforward.
Definition. The topology $\mathcal{T}_Y$ is known as the relative, subspace, or restricted topology on $Y$ induced by the topology $\mathcal{T}$.
Theorem 5.1.6. Let $A \subseteq Y$, and let $\bar{A}_Y$ denote the closure of $A$ in $\left(Y, \mathcal{T}_Y\right)$. Then $\bar{A}_Y=\bar{A} \cap Y$.
Proof. Since $\bar{A}$ is closed in $X, \bar{A} \cap Y$ is closed in $Y$. Since $A \subseteq \bar{A} \cap Y, \bar{A}_Y \subseteq \bar{A} \cap Y$. We prove the reverse containment. Since $\bar{A}_Y$ is closed in $Y$, there exists a closed subset $F$ of $X$ such that $\bar{A}_Y=F \cap Y$. Thus $F$ is a closed subset of $X$, and $A \subseteq F$. Hence $\bar{A} \subseteq F$, and $\bar{A} \cap Y \subseteq F \cap Y=\bar{A}_Y$

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Definitions and Basic Properties

(a) $\varnothing$和$X$分别代表$\mathcal{T}$;
(b) $\mathcal{T}$的任意成员族的并集是$\mathcal{T}$的成员，并且
(c) $\mathcal{T}$的两个元素的交集是$\mathcal{T}$的一个元素。

## 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。

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## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Continuity and Equivalent Metrics

Continuity, from the intuitive point of view, is about the gradual rather than the abrupt change of function values. In its simplest form, the graph of a continuous, real-valued function of a single real variable must be connected. Most functions in mathematics are too complicated for such a visual characterization of continuity, and a more rigorous and robust definition is needed. The $\epsilon-\delta$ definition of continuity revolutionized calculus, and hence mathematics, in the early nineteenth century. It is based on the idea that the fluctuations of a continuous function can be controlled in a sufficiently small neighborhood of a point of continuity. Our definition of local continuity in the metric setting is an immediate generalization of the $\epsilon-\delta$ definition. We then define the global continuity of a function on a metric space, an important concept seldom treated in undergraduate textbooks. You will see that continuity does not depend on the specific metric we use to measure proximity, but rather on the collection of open sets the metric induces. This leads us to the notion of equivalent metrics and, more generally, homeomorphisms.
Definition. Let $(X, d)$ and $(Y, \rho)$ be metric spaces. A function $f: X \rightarrow Y$ is said to be continuous at a point $x_0 \in X$ if, for every $\epsilon>0$, there exists $\delta>0$ such that $\rho\left(f(x), f\left(x_0\right)\right)<\epsilon$ whenever $d\left(x, x_0\right)<\delta$.
The following theorem is an obvious restatement of the definition.
Theorem 4.3.1. Let $(X, d)$ and $(Y, \rho)$ be metric spaces. A function $f: X \rightarrow Y$ is continuous at $x_0$ if and only if the inverse image of an open ball in $Y$ centered at $f\left(x_0\right)$ contains an open ball in $X$ centered at $x_0$.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Homeomorphisms

The concept of a homeomorphism is of central importance in topology. In the metric setting, isometry, although quite useful, is too stringent and does not preclude homeomorphisms from being useful. One can loosely think of a homeomorphism as a relaxation of the concept of isometry and an extension of the notion of metric equivalence.
Definition. Two metric spaces $(X, d)$ and $(Y, \rho)$ are homeomorphic if there exists a bicontinuous bijection $\varphi$ from $X$ to $Y$. The function $\varphi$ is called a homeomorphism from $X$ to $Y$.
Example 10. The open interval $(-1,1)$ is homeomorphic to $\mathbb{R}$ (both sets have the usual metric). The function $f(t)=\frac{t}{1-t^2} \operatorname{maps}(-1,1)$ bicontinuously onto $\mathbb{R}$.
Example 11. The closed upper half plane $H=\mathbb{R} \times[0, \infty)$ is homeomorphic to the half-open strip $A=\mathbb{R} \times[0,1)$. To see this, define $\varphi: H \rightarrow A$ by $\varphi(x, y)=$ $\left(x, \frac{y}{1+y}\right)$. It is a rather routine matter to verify that $\varphi$ is a bijection and that its inverse is $\varphi^{-1}(x, t)=\left(x, \frac{t}{1-t}\right)$.
Example 12 (the stereographic projection). Let $\mathcal{S}^1=\left{\left(\xi_1, \xi_2\right) \in \mathbb{R}^2: \xi_1^2+\xi_2^2-\right.$ $\left.\xi_2=0\right}$ be a circle of diameter 1 and centered at the point $(0,1 / 2)$, and let $N=$ $(0,1)$ be the top point on the circle. Define the punctured circle to be the circle with the top point removed: $\mathcal{S}^1=\mathcal{S}^1-{N}$. We give $\mathcal{S}^1$ the Euclidean metric in the plane. Define a bijection $P: \mathcal{S}^1 \rightarrow \mathbb{R}$ as follows: for a point $\xi=\left(\xi_1, \xi_2\right) \in \mathcal{S}^1$, $P(\xi)$ is the horizontal intercept of the line that contains the points $N$ and $\xi$, as shown in figure 4.1.

## 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。

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## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Definitions and Basic Properties

This section is a summary of the most basic concepts of vector space theory. The main reason for including this section is to establish terminology and provide a collection of important examples. The reader should pay particular attention to the examples, because the sequence and function spaces we introduce here are of fundamental importance for the rest of the book. The theorems are stated without proof.

Definition. Let $\mathbb{K}$ be a field, and suppose $U$ is a nonempty set equipped with a binary operation, + (vector addition). Suppose also that there is a function $\times: \mathbb{K} \times U \rightarrow U$ (scalar multiplication) that assigns to each pair $(a, u) \in \mathbb{K} \times U$ an element $a \times u$ (or simply $a u$ ) in $U$. The triple $(U,+, x)$ is called a vector space over the field $\mathbb{K}$ if the following conditions are satisfied by all elements $a, b \in \mathbb{K}$ and all elements $u, v, w \in U:$
(a) $u+v=v+u$
(b) $u+(v+w)=(u+v)+w$.
(c) There is an element $0 \in V$ (the zero vector) such that $u+0=u$.
(d) For every $u \in U$, there is an element $-u \in U$ such that $u+(-u)=0$.
(e) $a(u+v)=a u+a v$.
(f) $(a+b) u=a u+b u$.
(g) $(a b) u=a(b u)$.
(h) $1 . u=u$.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Independent Sets and Bases

This section is focused on the concepts on linear independence and bases. Our approach to studying bases is unified in the sense that we do not treat finitedimensional and infinite-dimensional spaces separately. We use Zorn’s lemma to prove the existence of a basis. A number of important equivalent characterizations of a basis are also discussed, both in the body of the section as well as in the section exercises.

Definition. A finite subset $\left{u_1, u_2, \ldots, u_n\right}$ of a vector space $U$ is dependent if there exist scalars $a_1, a_2, \ldots, a_n$, not all zero, such that $\sum_{i=1}^n a_i u_i=0$.

Terminology. A vector of the form $\sum_{i=1}^n a_i u_i$, where at least one $a_i \neq 0$, is called a nontrivial linear combination of $u_1, u_2, \ldots, u_n$. The above definition can be restated as follows: $\left{u_1, u_2, \ldots, u_n\right}$ is dependent if some nontrivial linear combination of $u_1, u_2, \ldots, u_n$ is zero.

Theorem 3.2.1. A subset $S=\left{u_1, u_2, \ldots, u_n\right}$ of a vector space $U$ is dependent if and only if one of the vectors in $S$ is a linear combination of the remaining vectors.
Proof. Suppose $\left{u_1, u_2, \ldots, u_n\right}$ is dependent. Then $\sum_{i=1}^n a_i u_i=0$ for scalars $a_1, a_2, \ldots, a_n$, not all zero. Say $a_i \neq 0$. Then $u_i=\frac{-1}{a_i} \sum_{j \neq i}^n a_j u_j$. Conversely, if $u_i=\sum_{j \neq i}^n a_j u_j$, then $1 . u_i-\sum_{j \neq i}^n a_j u_j=0$, and $\left{u_1, u_2, \ldots, u_n\right}$ is dependent.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Definitions and Basic Properties

(a) $u+v=v+u$
(b) $u+(v+w)=(u+v)+w$。
(c)有一个元素$0 \in V$(零向量)使得$u+0=u$。
(d)对于每一个$u \in U$，都有一个元素$-u \in U$，使得$u+(-u)=0$。
(e) $a(u+v)=a u+a v$。
(f) $(a+b) u=a u+b u$。
(g) $(a b) u=a(b u)$。
(h) $1 . u=u$。

## 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。