Pub Date : 2019-01-25DOI: 10.1201/9780203749807-16
Yau-chuen Wong
{"title":"Dual Pairs and the Weak Topology","authors":"Yau-chuen Wong","doi":"10.1201/9780203749807-16","DOIUrl":"https://doi.org/10.1201/9780203749807-16","url":null,"abstract":"","PeriodicalId":218491,"journal":{"name":"Introductory Theory of Topological Vector Spaces","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124245455","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":"A Structure Theorem for Compact Sets in a Banach Space","authors":"Yau-chuen Wong","doi":"10.1201/9780203749807-5","DOIUrl":"https://doi.org/10.1201/9780203749807-5","url":null,"abstract":"","PeriodicalId":218491,"journal":{"name":"Introductory Theory of Topological Vector Spaces","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123367434","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}
Pub Date : 2019-01-25DOI: 10.1201/9780203749807-10
Yau-chuen Wong
{"title":"Normed Spaces Associated with a Locally Convex Space","authors":"Yau-chuen Wong","doi":"10.1201/9780203749807-10","DOIUrl":"https://doi.org/10.1201/9780203749807-10","url":null,"abstract":"","PeriodicalId":218491,"journal":{"name":"Introductory Theory of Topological Vector Spaces","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123367649","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":"Projective Topologies and Inductive Topologies","authors":"Yau-chuen Wong","doi":"10.1201/9780203749807-9","DOIUrl":"https://doi.org/10.1201/9780203749807-9","url":null,"abstract":"","PeriodicalId":218491,"journal":{"name":"Introductory Theory of Topological Vector Spaces","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126250208","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}
Recall the definitions of a norm and inner product on a real vector space V. We will assume a basic knowledge of these concepts, as can be found in any linear algebra book. The following proposition defines a metric topology on any normed vector space: Then d is a metric on V , and the norm · is continuous in the metric topology. PROOF The axioms for a metric are easy to check — the triangle inequality follows from the triangle inequality for norms. The continuity of the norm follows from the continuity of d and the fact that v = d(v, 0). Definition: Banach Spaces A Banach space is a normed vector space whose associated metric is complete. For example, any norm on a finite-dimensional vector space is complete, and therefore any finite-dimensional normed vector space is a Banach space. However, the term " Banach space " is mostly used only in the context of infinite-dimensional spaces. We begin by giving some basic examples of infinite-dimensional Banach spaces.
{"title":"Banach Spaces and Hilbert Spaces","authors":"Yau-Chuen Wong","doi":"10.1201/9780203749807-2","DOIUrl":"https://doi.org/10.1201/9780203749807-2","url":null,"abstract":"Recall the definitions of a norm and inner product on a real vector space V. We will assume a basic knowledge of these concepts, as can be found in any linear algebra book. The following proposition defines a metric topology on any normed vector space: Then d is a metric on V , and the norm · is continuous in the metric topology. PROOF The axioms for a metric are easy to check — the triangle inequality follows from the triangle inequality for norms. The continuity of the norm follows from the continuity of d and the fact that v = d(v, 0). Definition: Banach Spaces A Banach space is a normed vector space whose associated metric is complete. For example, any norm on a finite-dimensional vector space is complete, and therefore any finite-dimensional normed vector space is a Banach space. However, the term \" Banach space \" is mostly used only in the context of infinite-dimensional spaces. We begin by giving some basic examples of infinite-dimensional Banach spaces.","PeriodicalId":218491,"journal":{"name":"Introductory Theory of Topological Vector Spaces","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127669115","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":"Operators and Some Consequences of Hahn-Banach’s Extension Theorem","authors":"Yau-chuen Wong","doi":"10.1201/9780203749807-3","DOIUrl":"https://doi.org/10.1201/9780203749807-3","url":null,"abstract":"","PeriodicalId":218491,"journal":{"name":"Introductory Theory of Topological Vector Spaces","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132819994","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}
Pub Date : 2019-01-25DOI: 10.1201/9780203749807-20
Yau-chuen Wong
{"title":"Precompact Seminorms and Schwartz Spaces","authors":"Yau-chuen Wong","doi":"10.1201/9780203749807-20","DOIUrl":"https://doi.org/10.1201/9780203749807-20","url":null,"abstract":"","PeriodicalId":218491,"journal":{"name":"Introductory Theory of Topological Vector Spaces","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130780448","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}
Pub Date : 2019-01-25DOI: 10.1201/9780203749807-14
Yau-chuen Wong
{"title":"Initial Bornologies and Final Bornologies","authors":"Yau-chuen Wong","doi":"10.1201/9780203749807-14","DOIUrl":"https://doi.org/10.1201/9780203749807-14","url":null,"abstract":"","PeriodicalId":218491,"journal":{"name":"Introductory Theory of Topological Vector Spaces","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127915356","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}
Pub Date : 2019-01-25DOI: 10.1201/9780203749807-19
Yau-chuen Wong
{"title":"Some Recent Results on Compact Operators and Weakly Compact Operators","authors":"Yau-chuen Wong","doi":"10.1201/9780203749807-19","DOIUrl":"https://doi.org/10.1201/9780203749807-19","url":null,"abstract":"","PeriodicalId":218491,"journal":{"name":"Introductory Theory of Topological Vector Spaces","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115913508","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}
Pub Date : 2019-01-25DOI: 10.1201/9780203749807-17
Yau-chuen Wong
{"title":"Elementary Duality Theory","authors":"Yau-chuen Wong","doi":"10.1201/9780203749807-17","DOIUrl":"https://doi.org/10.1201/9780203749807-17","url":null,"abstract":"","PeriodicalId":218491,"journal":{"name":"Introductory Theory of Topological Vector Spaces","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114769796","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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