Pub Date : 2020-03-12DOI: 10.1017/9781139030267.028
{"title":"Weak and Weak-* Convergence","authors":"","doi":"10.1017/9781139030267.028","DOIUrl":"https://doi.org/10.1017/9781139030267.028","url":null,"abstract":"","PeriodicalId":256579,"journal":{"name":"An Introduction to Functional Analysis","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114494188","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 : 2020-03-12DOI: 10.1017/9781139030267.003
Christian Clason
As calculus developed, eventually turning into analysis, concepts first explored on the real line (e.g., a limit of a sequence of real numbers) eventually extended to other spaces (e.g., a limit of a sequence of vectors or of functions), and in the early 20th century a general setting for analysis was formulated, called a metric space. It is a set on which a notion of distance between each pair of elements is defined, and in which notions from calculus in R (open and closed intervals, convergent sequences, continuous functions) can be studied. Many of the fundamental types of spaces used in analysis are metric spaces (e.g., Hilbert spaces and Banach spaces), so metric spaces are one of the first abstractions that has to be mastered in order to learn analysis.
{"title":"Metric Spaces","authors":"Christian Clason","doi":"10.1017/9781139030267.003","DOIUrl":"https://doi.org/10.1017/9781139030267.003","url":null,"abstract":"As calculus developed, eventually turning into analysis, concepts first explored on the real line (e.g., a limit of a sequence of real numbers) eventually extended to other spaces (e.g., a limit of a sequence of vectors or of functions), and in the early 20th century a general setting for analysis was formulated, called a metric space. It is a set on which a notion of distance between each pair of elements is defined, and in which notions from calculus in R (open and closed intervals, convergent sequences, continuous functions) can be studied. Many of the fundamental types of spaces used in analysis are metric spaces (e.g., Hilbert spaces and Banach spaces), so metric spaces are one of the first abstractions that has to be mastered in order to learn analysis.","PeriodicalId":256579,"journal":{"name":"An Introduction to Functional Analysis","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114272347","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 : 2020-03-12DOI: 10.1017/9781139030267.016
F. Bonsall
Proof. Since the dimensions of R(A) is always small or equal to the dimension of N (A)⊥, X and R(A), and N (A)⊥ are all infinite dimensional. Hence, we can find a sequence (xn) with xn ∈ N (A)⊥ with ‖xn‖ = 1 and 〈xn, xm〉 = 0 for n 6= m. Since A is compact, the sequence (yn) = (Axn) has to contain a convergent subsequence. Thus, for any δ > 0 we can find k and l such that ‖yk − yl‖ < δ. However, ‖A(yk − yl)‖ = ‖xk − xl‖ = ‖xk‖ + ‖xl‖ − 2〈xk, xl〉 = 2, although A†0 = 0. Hence, A† is not continuous.
{"title":"Compact Linear Operators","authors":"F. Bonsall","doi":"10.1017/9781139030267.016","DOIUrl":"https://doi.org/10.1017/9781139030267.016","url":null,"abstract":"Proof. Since the dimensions of R(A) is always small or equal to the dimension of N (A)⊥, X and R(A), and N (A)⊥ are all infinite dimensional. Hence, we can find a sequence (xn) with xn ∈ N (A)⊥ with ‖xn‖ = 1 and 〈xn, xm〉 = 0 for n 6= m. Since A is compact, the sequence (yn) = (Axn) has to contain a convergent subsequence. Thus, for any δ > 0 we can find k and l such that ‖yk − yl‖ < δ. However, ‖A(yk − yl)‖ = ‖xk − xl‖ = ‖xk‖ + ‖xl‖ − 2〈xk, xl〉 = 2, although A†0 = 0. Hence, A† is not continuous.","PeriodicalId":256579,"journal":{"name":"An Introduction to Functional Analysis","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129633041","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 : 2020-03-12DOI: 10.1017/9781139030267.010
{"title":"Orthonormal Sets and Orthonormal Bases for Hilbert Spaces","authors":"","doi":"10.1017/9781139030267.010","DOIUrl":"https://doi.org/10.1017/9781139030267.010","url":null,"abstract":"","PeriodicalId":256579,"journal":{"name":"An Introduction to Functional Analysis","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131606300","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 : 2020-03-12DOI: 10.1017/9781139030267.024
{"title":"The Open Mapping, Inverse Mapping, and Closed Graph Theorems","authors":"","doi":"10.1017/9781139030267.024","DOIUrl":"https://doi.org/10.1017/9781139030267.024","url":null,"abstract":"","PeriodicalId":256579,"journal":{"name":"An Introduction to Functional Analysis","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126356581","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 : 2020-03-12DOI: 10.1007/978-1-4757-1645-0_16
Paul R. Halmos
{"title":"Zorn’s Lemma","authors":"Paul R. Halmos","doi":"10.1007/978-1-4757-1645-0_16","DOIUrl":"https://doi.org/10.1007/978-1-4757-1645-0_16","url":null,"abstract":"","PeriodicalId":256579,"journal":{"name":"An Introduction to Functional Analysis","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115038275","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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