Megginson An Introduction To Banach Space Theory Pdf 12 __FULL_

The purpose of this book is to serve as a text for a graduatecourse in functional analysis emphasizing Banach space theory.Its intended audience is graduate students who have had the standardcourses in analysis and measure theory up to and including elementaryproperties of the p spaces, but who may not yethave seen any of the basic results from a first course in functionalanalysis, such as the uniform boundedness principle and the variousforms of the Hahn-Banach theorem. A detailed description of theprerequisites for reading this textis given below.

Though many of the classical results of functional analysisare presented in this text, these results are applied for themost part to normed spaces in general and Banach spaces in particular,allowing a much more extensive development of that theory whileplacing correspondingly less emphasis on other topics that mightappear in a traditional functional analysis text. For more informationon the contents of the text, see the Tableof Contents and section-by-sectionsynopsis below.

Megginson An Introduction To Banach Space Theory Pdf 12

Download 🔥 https://blltly.com/2y1FGD 🔥

The theory is developed for normed spaces over both the realand complex scalar fields. When a result holds for incompletenormed spaces as well as Banach spaces, the result is usuallystated and proved in the more general form so that the readerwill know where completeness is truly essential. However, resultsthat can be extended from Banach spaces or arbitrary normed spacesto larger classes of topological vector spaces usually do notget the more general treatment unless the extension has a specificapplication to Banach space theory.

This book is sprinkled liberally with examples, both to showthe theory at work and to illustrate why certain hypotheses intheorems are necessary. The book is also sprinkled liberally withhistorical notes and citations of original sources, with specialattention given to mentioning dates within the body of the textso that the reader can get a feeling for the time frame withinwhich the different parts of Banach space theory evolved.

Anyone who has studied the first third of Walter Rudin's Realand Complex Analysis, which is to say the first six chaptersof that book, will be able to read this book through, cover-to-cover,omitting nothing. Of course, this implies that the reader hashad the basic grounding in undergraduate mathematics necessaryto tackle Rudin's book, which should include a first course inlinear algebra. Though some knowledge of elementary topology beyondthe theory of metric spaces is assumed, the topology presentednear the beginning of Rudin's book is enough. In short, all ofthis book is accessible to someone who has had a course in realand complex analysis that includes the duality between the Lebesguespaces p and q when 1

Chapter 1 focuses on the metric theory of normed spaces.The first three sections present fundamental definitions and examples,as well as the most elementary properties of normed spaces suchas the continuity of their vector space operations. The fourthsection contains a short development of the most basic propertiesof bounded linear operators between normed spaces, including propertiesof normed space isomorphisms, which are then used to show thatevery finite-dimensional normed space is a Banach space.

In Section 1.7, the properties of quotient spaces formed fromnormed spaces are examined and the first isomorphism theorem forBanach spaces is proved: If T is a bounded linear operatorfrom a Banach space X onto a Banach space Y, thenY and X/ker(T) are isomorphic as Banach spaces.Following a section devoted to direct sums of normed spaces, Section1.9 presents the vector space and normed space versions of theHahn-Banach extension theorem, along with their close relative,Helly's theorem for bounded linear functionals. The same sectioncontains a development of Minkowski functionals and gives an exampleof how they are used to prove versions of the Hahn-Banach separationtheorem. Section 1.10 introduces the dual space of a normed space,and has the characterizations up to isometric isomorphism of theduals of direct sums, quotient spaces, and subspaces of normedspaces. The next section discusses reflexivity and includes Pettis'stheorem about the reflexivity of a closed subspace of a reflexivespace and many of its consequences. Section 1.12, devoted to separability,includes the Banach-Mazur characterization of separable Banachspaces as isomorphs of quotient spaces of \ell_1, and ends withthe characterization of separable normed spaces as the normedspaces that are compactly generated so that the stage is set forthe introduction of weakly compactly generated normed spaces inSection 2.8. This completes the basic material of Chapter 1.

The last section of Chapter 1, Section 1.13, is optional inthe sense that none of the material in the rest of the book outsideof other optional sections depends on it. This section containsa number of useful characterizations of reflexivity, includingJames's theorem. Some of the more basic of these are usually obtainedas corollaries of the Eberlein-Smulian theorem, but are includedhere since they can be proved fairly easily without it. The mostimportant of these basic characterizations are repeated in Section2.8 after the Eberlein-Smulian theorem is proved, so this sectioncan be skipped without fear of losing them. The heart of the sectionis a proof of the general case of James's theorem: A Banachspace is reflexive if each bounded linear functional x* on thespace has the property that the supremum of |x*| on the closedunit ball of the space is attained somewhere on that ball.The proof given here is a detailed version of James's 1972 proof.While the development leading up to the proof could be abbreviatedslightly by delaying this section until the Eberlein-Smulian theoremis available, there are two reasons for my not doing so. The firstis that I wish to emphasize that the proof is really based onlyon the elementary metric theory of Banach spaces, not on argumentsinvolving weak compactness, and the best way to do that is togive the proof before the weak topology has even been defined(though I do cheat a bit by defining weak sequential convergencewithout direct reference to the weak topology). The second isdue to the reputation that James's theorem has acquired as beingformidably deep. The proof is admittedly a bit intricate, butit is entirely elementary, not all that long, and contains somevery nice ideas. By placing the proof as early as possible inthis book, I hope to stress its elementary nature and dispel abit of the notion that it is inaccessible.

Chapter 2 deals with the weak topology of a normed spaceand the weak* topology of its dual. The first section includessome topological preliminaries, but is devoted primarily to afairly extensive development of the theory of nets, includingcharacterizations of topological properties in terms of the accumulationand convergence of certain nets. Even a student with a solid firstcourse in general topology may never have dealt with nets, soseveral examples are given to illustrate both their similaritiesto and differences from sequences. A motivation of the somewhatnonintuitive definition of a subnet is given, along with examples.The section includes a short discussion of topological groups,primarily to be able to obtain a characterization of relativecompactness in topological groups in terms of the accumulationof nets that does not always hold in arbitrary topological spaces.Ultranets are not discussed in this section, since they are notreally needed in the rest of this book, but a brief discussionof ultranets is given in Appendix D for use by the instructorwho wishes to show how ultranets can be used to simplify certaincompactness arguments.

Section 2.2 presents the basic properties of topological vectorspaces and locally convex spaces needed for a study of the weakand weak* topologies. The section includes a brief introductionto the dual space of a topological vector space, and presentsthe versions of the Hahn-Banach separation theorem due to Mazurand Eidelheit as well as the consequences for locally convex spacesof Mazur's separation theorem that parallel the consequences fornormed spaces of the normed space version of the Hahn-Banach extensiontheorem.

Chapter 3 contains a discussion of linear operatorsbetween normed spaces far more extensive than the brief introductionpresented in Section 1.4. The first section of the chapter isdevoted to adjoints of bounded linear operators between normedspaces. The second focuses on projections and complemented subspaces,and includes Whitley's short proof of Phillips's theorem that0 is not complemented in \ell_\infty.

There is another notion of completeness besides metric completeness and that is the notion of a complete topological vector space (TVS) or TVS-completeness, which uses the theory of uniform spaces. Specifically, the notion of TVS-completeness uses a unique translation-invariant uniformity, called the canonical uniformity, that depends only on vector subtraction and the topology {\displaystyle \tau } that the vector space is endowed with, and so in particular, this notion of TVS completeness is independent of whatever norm induced the topology {\displaystyle \tau } (and even applies to TVSs that are not even metrizable). Every Banach space is a complete TVS. Moreover, a normed space is a Banach space (that is, its norm-induced metric is complete) if and only if it is complete as a topological vector space. If ( X , ) {\displaystyle (X,\tau )} is a metrizable topological vector space (such as any norm induced topology, for example), then ( X , ) {\displaystyle (X,\tau )} is a complete TVS if and only if it is a sequentially complete TVS, meaning that it is enough to check that every Cauchy sequence in ( X , ) {\displaystyle (X,\tau )} converges in ( X , ) {\displaystyle (X,\tau )} to some point of X {\displaystyle X} (that is, there is no need to consider the more general notion of arbitrary Cauchy nets). be457b7860