LaTeX 第一篇文章

一、怎么建立和编译文档

二、编写

% Finite Dimensional Normed Linear Linear Space.tex

\documentclass{ctexart}

\usepackage{amsmath}

\usepackage{amssymb}

\title{Finite Dimensional Normed Linear Linear Space}

\author{杨书惠}

\begin{document}

\section{Theorem}

Let $(X,\Vert \cdot \Vert)$ be a finite-dimensional  normed linear space with basis ${ x_1 , x_2 ,\dots , x_n}$ .Then there is a constant $m>0$ such that for every choice of scalars $ \alpha_1 , \alpha_2 , \dots ,\alpha_n$ ,we have

\[

m\sum_{j=1}^n \vert \alpha_j \vert \le \lVert \sum_{j=1}^n \alpha_j x_j \rVert .

\]

\\\ \textbf{Proof.}

If $\sum_{j=1}^n \vert \alpha_j \vert =0 $ ,then $\alpha_j=0$ for all $j=1,2, \dots ,n$ and the inequality holds for any $m>0$.

\\\ Assume that $\sum_{j=1}^n \lvert \alpha_j \rvert \neq 0$ .We shall prove the result for a set of scalars $\lbrace \alpha_1 ,\alpha_2 ,\cdot , \alpha_n \rbrace$ that satisfy the condition $\sum_{j=1}^n \lvert \alpha_j \rvert=1$ . Let

\[

A=\lbrace (\alpha_1 , \alpha_2 , \dots , \alpha_n)\in \mathbb{F}_n \vert  \sum_{j=1}^n \lvert\alpha_j\rvert=1\rbrace .

\]

Since $A$ is a closed and bounded subset of $\mathbb{F}_n$ ,it is compact. Define $f:A\to\mathbb{R}$ by

\[

f(\alpha_1,\alpha_2,\dots,\alpha_n)=\lVert \sum_{j=1}^n\alpha_jx_j \rVert .

\]

\end{document}

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