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# 数学代写|拓扑学代写TOPOLOGY代考|MATH10076 THE HAHN-BANACH THEOREM

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## 数学代写|拓扑学代写TOPOLOGY代考|THE HAHN-BANACH THEOREM

One of the basic principles of strategy in the study of an abstract mathematical system can be stated as follows: consider the set of all structure-preserving mappings of that system into the simplest system of the same type. This principle is richly fruitful in the structure theory (or representation theory) of groups, rings, and algebras, and we shall see in the next section how it works for normed linear spaces.

We have remarked that the spaces $R$ and $C$ are the simplest of all normed linear spaces. If $N$ is an arbitrary normed linear space, the above principle leads us to form the set of all continuous linear transformations of $N$ into $R$ or $C$, according as $N$ is real or complex. This setit is $Q(N, R)$ or $Q(N, C)$-is denoted by $N^{}$ and is called the conjugate space of $N$. The elements of $N^{}$ are called continuous linear functionals, or more briefly, functionals. ${ }^{1}$ It follows from our work in the previous section that if these functionals are added and multiplied by scalars
${ }^{1}$ The noun “functional” seems to have originated in the theory of integral equations. It was used to distinguish between a function in the elementary sense defined on a set of numbers and a function (or functional) defined on a set of functions. In this book, we always use the word to mean a scalar-valued continuous linear function defined on a normed linear space.

## 数学代写|拓扑学代写TOPOLOGY代考|THE NATURAL IMBEDDING OF N IN N* *

Since the conjugate space $N^{}$ of a normed linear space $N$ is itself a normed linear space, it is possible to form the conjugate space $\left(N^{}\right)^{}$ of $N^{}$. We denote this space by $N^{* *}$, and we call it the second conjugate space of $N$.

The importance of $N^{* }$ rests on the fact that each vector $x$ in $N$ gives rise to a functional $F_{x}$ in $N^{ }$. If we denote a typical element of $N^{}$ by $f$, then $F_{x}$ is defined by
$$F_{x}(f)=f(x) .$$
In other words, we invert the usual practice by regarding the symbol $f(x)$ as specifying a function of $f$ for each fixed $x$, and we emphasize this point of view by writing $f(x)$ in the form $F_{x}(f)$. A simple manipulation of the definition shows that $F_{x}$ is linear:
\begin{aligned} F_{x}(\alpha f+\beta g) &=(\alpha f+\beta g)(x) \ &=\alpha f(x)+\beta g(x) \ &=\alpha F_{x}(f)+\beta F_{x}(g) \end{aligned}
If we now compute the norm of $F_{x}$, we see that
\begin{aligned} \left|F_{x}\right| &=\sup \left{\left|F_{x}(f)\right|:|f| \leq 1\right} \ &=\sup {|f(x)|:|f| \leq 1} \ & \leq \sup {|f||x|:|f| \leq 1} \ & \leq|x| \end{aligned}

## 数学代写|拓扑学代写TOPOLOGY代考|THE HAHN-BANACH THEOREM

$1^{1}$ 名词“泛函”似平起源于积分方程理论。它用于区分在一组数字上定义的基本意义上的函数和在一组函数上定义 的函数 (或泛函) 。在本书中，我们总是使用这个词来表示定义在规范线性空间上的标量值连续线性函数。

## 数学代写拓扑学代写TOPOLOGY代考|THE NATURAL IMBEDDING OF N IN N* *

$$F_{x}(f)=f(x) .$$

$$F_{x}(\alpha f+\beta g)=(\alpha f+\beta g)(x) \quad=\alpha f(x)+\beta g(x)=\alpha F_{x}(f)+\beta F_{x}(g)$$

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## 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。