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# 数学代写|数学分析作业代写Mathematical Analysis代考|MTH131 Vector Spaces

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## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|Vector Spaces

Peano was born in a farmhouse about $5 \mathrm{~km}$ from Cuneo, where he received his early education. One of Peano’s uncles was a priest and a lawyer in Turin, and he realized the child’s talent. He took him to Turin in 1870 for his secondary schooling. Peano entered the University of Turin in 1876, graduated in 1880 doctor of mathematics, and was appointed to the university the same year. $\mathrm{He}$ received his qualification to be a university professor in 1884 .

In 1886 Peano proved the existence of the solution of the differential equation $d y / d x=f(x, y)$ under the mere assumption that $f$ is continuous in the neighborhood of the initial point $\left(x_{0}, y_{0}\right)$. In 1888 he published the book Geometrical Calculus, which begins with a chapter on mathematical logic. A significant feature of the book is that, in it, Peano sets out with great clarity the ideas of Grassmann, who made the first attempt to define a vector space, albeit in a rather obscure way.

This book contains the first definition of a vector space given with a remarkably modern notation and style. This was, without a doubt, a big development in the history of mathematics. In 1889 Peano published an axiomatic approach to the definition of the natural numbers that was based on the notion of the successor function. In 1890 he made the stunning discovery that there are continuous surjective mappings from $[0,1]$ onto the unit square, which came to be known as space-filling curves.

Peano’s career was strangely divided into two periods. The period up to 1900 is one where he showed great originality and a remarkable feel for topics that would be important in the development of mathematics. His achievements were outstanding, and he had a modern style quite ahead of his own time. However, this feel for what was important seemed to leave him, and, after 1900, he worked with great enthusiasm on two projects of great difficulty, which were enormous undertakings but proved quite unimportant in the development of mathematics.

## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|The Evolution of the Concept of a Vector Space2

The emergence of the modern definition of a vector space was delayed for a considerable length of time because of several reasons. It appears that early attempts to define what we know now as a vector space were hindered by the insistence on incorporating axioms for determinants. The lack of awareness of the importance of axiomatics and abstract thinking was also a major obstacle. Grassmann’s pioneering ideas were obscured by philosophical language, and although Peano’s definition was a long step toward axiomatization, it did not produce the modern definition. The founders of functional analysis were instrumental in framing the modern definition. In 1916 Riesz studied spaces of continuous functions and defined linear transformations and even the concept of bounded linear operators. Decisive steps toward axiomatization were taken independently by Banach in 1920 and by Hahn, in two papers published in 1922 and 1927. In 1920 Banach took Riesz’s ideas one step further and defined what is known in modern terminology as a Banach space. The function spaces Banach and Riesz studied are infinite dimensional, and this makes the use of an axiomatic approach compulsory. Banach’s approach was confined to function spaces, and his axioms did not coincide with the modern definition in that some axioms are redundant and some are missing. Modern algebra finally paved the way toward the modern definition of a vector space: determinants were dropped from the axiomatic approach, and this unified the definition of finite and infinite-dimensional spaces. The definition was made accessible to beginning students in books that were published in the 1940 s by Birkhoff and MacLane, Halmos, and Bourbaki.

## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|Vector Spaces

1886年Peano证明了微分方程解的存在性d是/dX=F(X,是)仅假设F在初始点附近是连续的(X0,是0). 1888 年，他出版了《几何微积分》一书，该书以数学逻辑一章开头。这本书的一个显着特点是，皮亚诺在其中非常清晰地阐述了格拉斯曼的想法，他第一次尝试定义向量空间，尽管是以一种相当模糊的方式。

## 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。