{"title":"The Transpose (or Dual)of a Linear Operator","authors":"","doi":"10.1002/9781119851318.app2","DOIUrl":"https://doi.org/10.1002/9781119851318.app2","url":null,"abstract":"","PeriodicalId":233638,"journal":{"name":"From Euclidean to Hilbert Spaces","volume":"686 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129069424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Banach Spaces and Hilbert Spaces","authors":"F. Narcowich","doi":"10.1002/9781119851318.ch4","DOIUrl":"https://doi.org/10.1002/9781119851318.ch4","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":0.0,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134236496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Other titles from in Mathematics and Statistics","authors":"","doi":"10.1002/9781119851318.oth","DOIUrl":"https://doi.org/10.1002/9781119851318.oth","url":null,"abstract":"","PeriodicalId":233638,"journal":{"name":"From Euclidean to Hilbert Spaces","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127610185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Discrete Fourier Transform and its Applications to Signal and Image Processing","authors":"","doi":"10.1002/9781119851318.ch2","DOIUrl":"https://doi.org/10.1002/9781119851318.ch2","url":null,"abstract":"","PeriodicalId":233638,"journal":{"name":"From Euclidean to Hilbert Spaces","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133891676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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