{"title":"巴拿赫空间和希尔伯特空间","authors":"F. Narcowich","doi":"10.1002/9781119851318.ch4","DOIUrl":null,"url":null,"abstract":"A sequence {vj} is said to be Cauchy if for each > 0, there exists a natural number N such that ‖vj−vk‖ < for all j, k ≥ N . Every convergent sequence is Cauchy, but there are many examples of normed linear spaces V for which there exist non-convergent Cauchy sequences. One such example is the set of rational numbers Q. The sequence (1.4, 1.41, 1.414, . . . ) converges to √ 2 which is not a rational number. We say a normed linear space is complete if every Cauchy sequence is convergent in the space. The real numbers are an example of a complete normed linear space. We say that a normed linear space is a Banach space if it is complete. We call a complete inner product space a Hilbert space. Consider the following examples: 1. Every finite dimensional normed linear space is a Banach space. Likewise, every finite dimensional inner product space is a Hilbert space.","PeriodicalId":233638,"journal":{"name":"From Euclidean to Hilbert Spaces","volume":"67 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Banach Spaces and Hilbert Spaces\",\"authors\":\"F. Narcowich\",\"doi\":\"10.1002/9781119851318.ch4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A sequence {vj} is said to be Cauchy if for each > 0, there exists a natural number N such that ‖vj−vk‖ < for all j, k ≥ N . Every convergent sequence is Cauchy, but there are many examples of normed linear spaces V for which there exist non-convergent Cauchy sequences. One such example is the set of rational numbers Q. The sequence (1.4, 1.41, 1.414, . . . ) converges to √ 2 which is not a rational number. We say a normed linear space is complete if every Cauchy sequence is convergent in the space. The real numbers are an example of a complete normed linear space. We say that a normed linear space is a Banach space if it is complete. We call a complete inner product space a Hilbert space. Consider the following examples: 1. Every finite dimensional normed linear space is a Banach space. Likewise, every finite dimensional inner product space is a Hilbert space.\",\"PeriodicalId\":233638,\"journal\":{\"name\":\"From Euclidean to Hilbert Spaces\",\"volume\":\"67 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-08-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"From Euclidean to Hilbert Spaces\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1002/9781119851318.ch4\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"From Euclidean to Hilbert Spaces","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1002/9781119851318.ch4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A sequence {vj} is said to be Cauchy if for each > 0, there exists a natural number N such that ‖vj−vk‖ < for all j, k ≥ N . Every convergent sequence is Cauchy, but there are many examples of normed linear spaces V for which there exist non-convergent Cauchy sequences. One such example is the set of rational numbers Q. The sequence (1.4, 1.41, 1.414, . . . ) converges to √ 2 which is not a rational number. We say a normed linear space is complete if every Cauchy sequence is convergent in the space. The real numbers are an example of a complete normed linear space. We say that a normed linear space is a Banach space if it is complete. We call a complete inner product space a Hilbert space. Consider the following examples: 1. Every finite dimensional normed linear space is a Banach space. Likewise, every finite dimensional inner product space is a Hilbert space.