Pub Date : 2020-03-12DOI: 10.1017/9781139030267.020
Christian Clason
The treatment given here is adapted from the third edition of Royden's Real Analysis (MacMillan, New York, 1988) and from the first few pages of Volume I of " Fundamentals of the Theory of Operator Algebras " Let V be a vector space over the field R of real numbers. x ± T for {x} ± T , etc.
这里给出的处理改编自Royden's Real Analysis (MacMillan, New York, 1988)的第三版和“算子代数理论基础”第一卷的前几页，设V是实数域R上的向量空间。为{x}±T的x±T，等等。
{"title":"The Hahn–Banach Theorem","authors":"Christian Clason","doi":"10.1017/9781139030267.020","DOIUrl":"https://doi.org/10.1017/9781139030267.020","url":null,"abstract":"The treatment given here is adapted from the third edition of Royden's Real Analysis (MacMillan, New York, 1988) and from the first few pages of Volume I of \" Fundamentals of the Theory of Operator Algebras \" Let V be a vector space over the field R of real numbers. x ± T for {x} ± T , etc.","PeriodicalId":256579,"journal":{"name":"An Introduction to Functional Analysis","volume":"11 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":"122632338","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: 10.1017/9781139030267.004
{"title":"Norms and Normed Spaces","authors":"","doi":"10.1017/9781139030267.004","DOIUrl":"https://doi.org/10.1017/9781139030267.004","url":null,"abstract":"","PeriodicalId":256579,"journal":{"name":"An Introduction to Functional Analysis","volume":"58 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":"127578223","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: 10.1017/9781108571814.002
A. DeCelles
Let V be a Hilbert space and D a subspace. A linear map T : D → V called an unbounded operator on V . This terminology is misleading since T is not necessarily defined on all of V and T may or may not be bounded. We denote the domain, D, of T as Dom(T ). Specifying the domain is an essential part of defining an unbounded operator. Often, but not always, the domain D is chosen to be dense in V . Usually, T is not continuous when D is given the subspace topology from V .
{"title":"Unbounded Operators on Hilbert Spaces","authors":"A. DeCelles","doi":"10.1017/9781108571814.002","DOIUrl":"https://doi.org/10.1017/9781108571814.002","url":null,"abstract":"Let V be a Hilbert space and D a subspace. A linear map T : D → V called an unbounded operator on V . This terminology is misleading since T is not necessarily defined on all of V and T may or may not be bounded. We denote the domain, D, of T as Dom(T ). Specifying the domain is an essential part of defining an unbounded operator. Often, but not always, the domain D is chosen to be dense in V . Usually, T is not continuous when D is given the subspace topology from V .","PeriodicalId":256579,"journal":{"name":"An Introduction to Functional Analysis","volume":"17 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":"126368006","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}
Our previous discussions have been concerned with algebra. The representation of systems (quantities and their interrelations) by abstract symbols has forced us to distill out the most significant and fundamental properties of these systems. We have been able to carry our exploration much deeper for linear systems, in most cases decomposing the system models into sets of uncoupled scalar equations. Our attention now turns to the geometric notions of length and angle. These concepts, which are fundamental to measurement and comparison of vectors, complete the analogy between general vector spaces and the physical three-dimensional space with which we are familiar. Then our intuition concerning the size and shape of objects provides us with valuable insight. The definition of length gives rigorous meaning to our previous heuristic discussions of an infinite sequence of vectors as a basis for an infinite-dimensional space. Length is also one of the most widely used optimization criteria. We explore this application of the concept of length in Chapter 6. The definition of orthogonality (or angle) allows us to carry even further our discussion of system decomposition. To this point, determination of the coordinates of a vector relative to a particular basis has required solution of a set of simultaneous equations. With orthogonal bases, each coordinate can be obtained independently, a much simpler process conceptually and, in some instances, computationally.
{"title":"Hilbert Spaces","authors":"Ryan Corning","doi":"10.1090/gsm/157/02","DOIUrl":"https://doi.org/10.1090/gsm/157/02","url":null,"abstract":"Our previous discussions have been concerned with algebra. The representation of systems (quantities and their interrelations) by abstract symbols has forced us to distill out the most significant and fundamental properties of these systems. We have been able to carry our exploration much deeper for linear systems, in most cases decomposing the system models into sets of uncoupled scalar equations. Our attention now turns to the geometric notions of length and angle. These concepts, which are fundamental to measurement and comparison of vectors, complete the analogy between general vector spaces and the physical three-dimensional space with which we are familiar. Then our intuition concerning the size and shape of objects provides us with valuable insight. The definition of length gives rigorous meaning to our previous heuristic discussions of an infinite sequence of vectors as a basis for an infinite-dimensional space. Length is also one of the most widely used optimization criteria. We explore this application of the concept of length in Chapter 6. The definition of orthogonality (or angle) allows us to carry even further our discussion of system decomposition. To this point, determination of the coordinates of a vector relative to a particular basis has required solution of a set of simultaneous equations. With orthogonal bases, each coordinate can be obtained independently, a much simpler process conceptually and, in some instances, computationally.","PeriodicalId":256579,"journal":{"name":"An Introduction to Functional Analysis","volume":" 22","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141223485","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: 10.1017/9781139030267.023
Many of the most important theorems in analysis assert that pointwise hypotheses imply uniform conclusions. Perhaps the simplest example is the theorem that a continuous function on a compact set is uniformly continuous. The main theorem in this section concerns a family of bounded linear operators, and asserts that the family is uniformly bounded (and hence equicontinuous) if it is pointwise bounded. We begin by defining these terms precisely.
{"title":"The Principle of Uniform Boundedness","authors":"","doi":"10.1017/9781139030267.023","DOIUrl":"https://doi.org/10.1017/9781139030267.023","url":null,"abstract":"Many of the most important theorems in analysis assert that pointwise hypotheses imply uniform conclusions. Perhaps the simplest example is the theorem that a continuous function on a compact set is uniformly continuous. The main theorem in this section concerns a family of bounded linear operators, and asserts that the family is uniformly bounded (and hence equicontinuous) if it is pointwise bounded. We begin by defining these terms precisely.","PeriodicalId":256579,"journal":{"name":"An Introduction to Functional Analysis","volume":"40 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":"129679745","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: 10.1017/9781139030267.013
{"title":"Dual Spaces and the Riesz Representation Theorem","authors":"","doi":"10.1017/9781139030267.013","DOIUrl":"https://doi.org/10.1017/9781139030267.013","url":null,"abstract":"","PeriodicalId":256579,"journal":{"name":"An Introduction to Functional Analysis","volume":"101 44","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131914351","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: 10.1017/9781139030267.025
{"title":"Spectral Theory for Compact Operators","authors":"","doi":"10.1017/9781139030267.025","DOIUrl":"https://doi.org/10.1017/9781139030267.025","url":null,"abstract":"","PeriodicalId":256579,"journal":{"name":"An Introduction to Functional Analysis","volume":"39 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":"133117198","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: 10.1017/9781139030267.015
{"title":"The Spectrum of a Bounded Linear Operator","authors":"","doi":"10.1017/9781139030267.015","DOIUrl":"https://doi.org/10.1017/9781139030267.015","url":null,"abstract":"","PeriodicalId":256579,"journal":{"name":"An Introduction to Functional Analysis","volume":"94 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":"114409352","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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