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全体正整数的自然对数构成N上的Hilbert空间

发表时间:2008-03-15  浏览量:1972  下载量:784
全部作者: 李汉巨
作者单位: 华南师范大学数学科学学院
摘 要: 本文引进自然数集上的线性空间的概念,并说明了全体正整数的自然对数构成的集合是上的线性空间。赋予一个内积,并证明了这个内积是完备的,即是上的Hilbert空间。进一步,研究了Hilbert空间的几何学。特别地,研究了的Gram-Schmidt正交化过程,得到了能够对线性无关向量组施行Gram-Schmidt正交化过程的充分必要条件,并利用这个结果证明了全体素数的自然对数构成的集合是的唯一正交基。最后,本文给出了此空间理论在数论中的应用。首先利用本文的方法证明了初等数论中的一些经典结果和两类数是无理数。其次研究了正整数集的几何学,定义了两个正整数的夹角和正整数集的极大集等概念,更为重要的是,证明了极大集的存在性,并提出了一些关于极大集的公开问题。
关 键 词: 数论;素数;极大集;标准正交基;Hilbert空间;Gram-Schmidt正交化
Title: The Hilbert space overconstructed by the natural logarithms of all integers
Author: LI Hanju
Organization: School of Mathematical Sciences, South China Normal University
Abstract: This paper introduces the notion of linear space over the natural numbersand shows that the set of natural logarithms of all ingegers is a linear space over. We endowwith an inner product and show thatis a Hilbert space over. Further, we study the geometry of Hilbert space. In particular, we study the process of Gram-Schmidt orthogonalization, and obtain the necessary and sufficient conditions for a collection of linear independent vectors to be able to operate the process of Gram-Schmidt orthogonalization. Using this result, we prove that the set of natural logarithms of all primes is the unique orthogonal basis of. Lastly, the paper applies this space theory to number theory. First, applying the method in the present paper, we prove some classic results in fundamental number theory and prove that two classes of number are irrational numbers. Secondly, we investigate the geometry of positive integers and define the notions of angle of two positive integers and maximum set of positive integers. More important, we prove the existentiality of maximum set and present some open problems about maximum set.
Key words: number theory; prime; maximum set; orthonormal basis; Hilbert space; Gram-Schmidt orthogonalization
发表期数: 2008年7月第5期
引用格式: 李汉巨. 全体正整数的自然对数构成N上的Hilbert空间[J]. 中国科技论文在线精品论文,2008,1(5):551-560.
 
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