Topological Vector Spaces and Their Applications Springer Monographs in Mathematics
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Topological Vector Spaces and Algebras
Joseph Muscat 2015 1 Topological Vector Spaces and Algebras [email protected] 1 June 2016 1 Topological Vector Spaces over R or C Recall that a topological vector space is a vector space with a T0 topology such that addition and the field action are continuous. When the field is F := R or C, the field action is called scalar multiplication. Examples: A N • R , such as sequences R , with pointwise convergence. p • Sequence spaces ℓ (real or complex) with topology generated by Br = (a ): p a p < r , where p> 0. { n n | n| } p p p p • LebesgueP spaces L (A) with Br = f : A F, measurable, f < r (p> 0). { → | | } R p • Products and quotients by closed subspaces are again topological vector spaces. If π : Y X are linear maps, then the vector space Y with the ini- i → i tial topology is a topological vector space, which is T0 when the πi are collectively 1-1. The set of (continuous linear) morphisms is denoted by B(X, Y ). The mor- phisms B(X, F) are called ‘functionals’. +, , Finitely- Locally Bounded First ∗ → Generated Separable countable Top. Vec. Spaces ///// Lp 0 <p< 1 ℓp[0, 1] (ℓp)N (ℓp)R p ∞ N n R 2 Locally Convex ///// L p > 1 L R , C(R ) R pointwise, ℓweak Inner Product ///// L2 ℓ2[0, 1] ///// ///// Locally Compact Rn ///// ///// ///// ///// 1. A set is balanced when λ 6 1 λA A. | | ⇒ ⊆ (a) The image and pre-image of balanced sets are balanced. ◦ (b) The closure and interior are again balanced (if A 0; since λA = (λA)◦ A◦); as are the union, intersection, sum,∈ scaling, T and prod- uct A ⊆B of balanced sets. -
Functional Analysis 1 Winter Semester 2013-14
Functional analysis 1 Winter semester 2013-14 1. Topological vector spaces Basic notions. Notation. (a) The symbol F stands for the set of all reals or for the set of all complex numbers. (b) Let (X; τ) be a topological space and x 2 X. An open set G containing x is called neigh- borhood of x. We denote τ(x) = fG 2 τ; x 2 Gg. Definition. Suppose that τ is a topology on a vector space X over F such that • (X; τ) is T1, i.e., fxg is a closed set for every x 2 X, and • the vector space operations are continuous with respect to τ, i.e., +: X × X ! X and ·: F × X ! X are continuous. Under these conditions, τ is said to be a vector topology on X and (X; +; ·; τ) is a topological vector space (TVS). Remark. Let X be a TVS. (a) For every a 2 X the mapping x 7! x + a is a homeomorphism of X onto X. (b) For every λ 2 F n f0g the mapping x 7! λx is a homeomorphism of X onto X. Definition. Let X be a vector space over F. We say that A ⊂ X is • balanced if for every α 2 F, jαj ≤ 1, we have αA ⊂ A, • absorbing if for every x 2 X there exists t 2 R; t > 0; such that x 2 tA, • symmetric if A = −A. Definition. Let X be a TVS and A ⊂ X. We say that A is bounded if for every V 2 τ(0) there exists s > 0 such that for every t > s we have A ⊂ tV . -
Order-Quasiultrabarrelled Vector Lattices and Spaces
Periodica Mathematica Hungarica YoL 6 (4), (1975), pp. 363--371. ~ ORDER-QUASIULTRABARRELLED VECTOR LATTICES AND SPACES by T. HUSAIN and S. M. KHALEELULLA (Hamilton) 1. Introduetion In this paper, we introduce and study a class of topological vector lattices (more generally, ordered topological vector spaces) which we call the class of order-quasiultrabarreUed vector lattices abbreviated to O. Q. U. vector lattices (respectively, O. Q. U. spaces). This class replaces that of order-infrabarrened Riesz spaces (respectively, order-infrabarrelled spaces) [8] in situations where local convexity is not assumed. We obtain an analogue of the Banach--Stein- haus theorem for lattice homom0rphisms on O. Q. U. vector lattices (respec- tively positive linear maps on O. Q. U. spaces) and the one for O. Q. U. spaces is s used successfully to obtain an analogue of the isomorphism theorem concerning O. Q.U. spaces and similar Schauder bases. Finally, we prove a closed graph theorem for O. Q. U, spaces, analogous to that for ultrabarrelled spa~s [6]. 2. Notations and preliminaries We abbreviate locally convex space, locally convex vector lattice, tope- logical vector space, and topological vector lattice to 1.c.s., 1.c.v.l., t.v.s., and t.v.l., respectively. We write (E, C) to denote a vector lattice (or ordered vector space) over the field of reals, with positive cone (or simply, cone) C. A subset B of a vector lattice (E, G) is solid if Ixl ~ ]y], yEB implies xEB; a vector subspace M of (~, C) which is also solid is called a lattice ideal. -
Some Special Sets in an Exponential Vector Space
Some Special Sets in an Exponential Vector Space Priti Sharma∗and Sandip Jana† Abstract In this paper, we have studied ‘absorbing’ and ‘balanced’ sets in an Exponential Vector Space (evs in short) over the field K of real or complex. These sets play pivotal role to describe several aspects of a topological evs. We have characterised a local base at the additive identity in terms of balanced and absorbing sets in a topological evs over the field K. Also, we have found a sufficient condition under which an evs can be topologised to form a topological evs. Next, we have introduced the concept of ‘bounded sets’ in a topological evs over the field K and characterised them with the help of balanced sets. Also we have shown that compactness implies boundedness of a set in a topological evs. In the last section we have introduced the concept of ‘radial’ evs which characterises an evs over the field K up to order-isomorphism. Also, we have shown that every topological evs is radial. Further, it has been shown that “the usual subspace topology is the finest topology with respect to which [0, ∞) forms a topological evs over the field K”. AMS Subject Classification : 46A99, 06F99. Key words : topological exponential vector space, order-isomorphism, absorbing set, balanced set, bounded set, radial evs. 1 Introduction Exponential vector space is a new algebraic structure consisting of a semigroup, a scalar arXiv:2006.03544v1 [math.FA] 5 Jun 2020 multiplication and a compatible partial order which can be thought of as an algebraic axiomatisation of hyperspace in topology; in fact, the hyperspace C(X ) consisting of all non-empty compact subsets of a Hausdörff topological vector space X was the motivating example for the introduction of this new structure. -
Spectral Radii of Bounded Operators on Topological Vector Spaces
SPECTRAL RADII OF BOUNDED OPERATORS ON TOPOLOGICAL VECTOR SPACES VLADIMIR G. TROITSKY Abstract. In this paper we develop a version of spectral theory for bounded linear operators on topological vector spaces. We show that the Gelfand formula for spectral radius and Neumann series can still be naturally interpreted for operators on topological vector spaces. Of course, the resulting theory has many similarities to the conventional spectral theory of bounded operators on Banach spaces, though there are several im- portant differences. The main difference is that an operator on a topological vector space has several spectra and several spectral radii, which fit a well-organized pattern. 0. Introduction The spectral radius of a bounded linear operator T on a Banach space is defined by the Gelfand formula r(T ) = lim n T n . It is well known that r(T ) equals the n k k actual radius of the spectrum σ(T ) p= sup λ : λ σ(T ) . Further, it is known −1 {| | ∈ } ∞ T i that the resolvent R =(λI T ) is given by the Neumann series +1 whenever λ − i=0 λi λ > r(T ). It is natural to ask if similar results are valid in a moreP general setting, | | e.g., for a bounded linear operator on an arbitrary topological vector space. The author arrived to these questions when generalizing some results on Invariant Subspace Problem from Banach lattices to ordered topological vector spaces. One major difficulty is that it is not clear which class of operators should be considered, because there are several non-equivalent ways of defining bounded operators on topological vector spaces. -
Locally Convex Spaces Manv 250067-1, 5 Ects, Summer Term 2017 Sr 10, Fr
LOCALLY CONVEX SPACES MANV 250067-1, 5 ECTS, SUMMER TERM 2017 SR 10, FR. 13:15{15:30 EDUARD A. NIGSCH These lecture notes were developed for the topics course locally convex spaces held at the University of Vienna in summer term 2017. Prerequisites consist of general topology and linear algebra. Some background in functional analysis will be helpful but not strictly mandatory. This course aims at an early and thorough development of duality theory for locally convex spaces, which allows for the systematic treatment of the most important classes of locally convex spaces. Further topics will be treated according to available time as well as the interests of the students. Thanks for corrections of some typos go out to Benedict Schinnerl. 1 [git] • 14c91a2 (2017-10-30) LOCALLY CONVEX SPACES 2 Contents 1. Introduction3 2. Topological vector spaces4 3. Locally convex spaces7 4. Completeness 11 5. Bounded sets, normability, metrizability 16 6. Products, subspaces, direct sums and quotients 18 7. Projective and inductive limits 24 8. Finite-dimensional and locally compact TVS 28 9. The theorem of Hahn-Banach 29 10. Dual Pairings 34 11. Polarity 36 12. S-topologies 38 13. The Mackey Topology 41 14. Barrelled spaces 45 15. Bornological Spaces 47 16. Reflexivity 48 17. Montel spaces 50 18. The transpose of a linear map 52 19. Topological tensor products 53 References 66 [git] • 14c91a2 (2017-10-30) LOCALLY CONVEX SPACES 3 1. Introduction These lecture notes are roughly based on the following texts that contain the standard material on locally convex spaces as well as more advanced topics. -
Functional Properties of Hörmander's Space of Distributions Having A
Functional properties of Hörmander’s space of distributions having a specified wavefront set Yoann Dabrowski, Christian Brouder To cite this version: Yoann Dabrowski, Christian Brouder. Functional properties of Hörmander’s space of distributions having a specified wavefront set. 2014. hal-00850192v2 HAL Id: hal-00850192 https://hal.archives-ouvertes.fr/hal-00850192v2 Preprint submitted on 3 May 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Communications in Mathematical Physics manuscript No. (will be inserted by the editor) Functional properties of H¨ormander’s space of distributions having a specified wavefront set Yoann Dabrowski1, Christian Brouder2 1 Institut Camille Jordan UMR 5208, Universit´ede Lyon, Universit´eLyon 1, 43 bd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France 2 Institut de Min´eralogie, de Physique des Mat´eriaux et de Cosmochimie, Sorbonne Univer- sit´es, UMR CNRS 7590, UPMC Univ. Paris 06, Mus´eum National d’Histoire Naturelle, IRD UMR 206, 4 place Jussieu, F-75005 Paris, France. Received: date / Accepted: date ′ Abstract: The space Γ of distributions having their wavefront sets in a closed cone Γ has become importantD in physics because of its role in the formulation of quantum field theory in curved spacetime. -
Arxiv:2105.06358V1 [Math.FA] 13 May 2021 Xml,I 2 H.3,P 31.I 7,Qudfie H Ocp Fa of Concept the [7]) Defined in Qiu Complete [7], Quasi-Fast for in As See (Denoted 1371]
INDUCTIVE LIMITS OF QUASI LOCALLY BAIRE SPACES THOMAS E. GILSDORF Department of Mathematics Central Michigan University Mt. Pleasant, MI 48859 USA [email protected] May 14, 2021 Abstract. Quasi-locally complete locally convex spaces are general- ized to quasi-locally Baire locally convex spaces. It is shown that an inductive limit of strictly webbed spaces is regular if it is quasi-locally Baire. This extends Qiu’s theorem on regularity. Additionally, if each step is strictly webbed and quasi- locally Baire, then the inductive limit is quasi-locally Baire if it is regular. Distinguishing examples are pro- vided. 2020 Mathematics Subject Classification: Primary 46A13; Sec- ondary 46A30, 46A03. Keywords: Quasi locally complete, quasi-locally Baire, inductive limit. arXiv:2105.06358v1 [math.FA] 13 May 2021 1. Introduction and notation. Inductive limits of locally convex spaces have been studied in detail over many years. Such study includes properties that would imply reg- ularity, that is, when every bounded subset in the in the inductive limit is contained in and bounded in one of the steps. An excellent introduc- tion to the theory of locally convex inductive limits, including regularity properties, can be found in [1]. Nevertheless, determining whether or not an inductive limit is regular remains important, as one can see for example, in [2, Thm. 34, p. 1371]. In [7], Qiu defined the concept of a quasi-locally complete space (denoted as quasi-fast complete in [7]), in 1 2 THOMASE.GILSDORF which each bounded set is contained in abounded set that is a Banach disk in a coarser locally convex topology, and proves that if an induc- tive limit of strictly webbed spaces is quasi-locally complete, then it is regular. -
Mackey +0-Barrelled Spaces Stephen A
Advances in Mathematics 145, 230238 (1999) Article ID aima.1998.1815, available online at http:ÂÂwww.idealibrary.com on Mackey +0-Barrelled Spaces Stephen A. Saxon* Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611-8105 E-mail: saxonÄmath.ufl.edu and Ian Tweddle* Department of Mathematics, University of Strathclyde, Glasgow G11XH, Scotland, United Kingdom E-mail: i.tweddleÄstrath.ac.uk CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - PublisherReceived Connector March 25, 1998; accepted December 14, 1998 In the context of ``Reinventing weak barrelledness,'' the best possible versions of the RobertsonSaxonRobertson Splitting Theorem and the SaxonTweddle Fit and Flat Components Theorem are obtained by weakening the hypothesis from ``barrelled'' to ``Mackey and +0-barreled.'' An example showing that the latter does not imply the former validates novelty, answers an old question, and completes a robust linear picture of ``Mackey weak barrelledness'' begun several decades ago. 1999 Academic Press Key Words: weak barrelledness; Mackey topology; splitting theorem. 1. INTRODUCTION Barrelled spaces have occupied an important place in the theory of locally convex spaces since its earliest days. This is probably due to the fact that they provide a vehicle for generalizing certain important properties of Banach spaces; for example, they form the class of domain spaces for a natural generalization of Banach's closed graph theorem and, in their dual characterization, they embody the conclusion of the BanachSteinhaus theorem for continuous linear functionals. Unlike Banach spaces, they are closed under the taking of inductive limits, countable-codimensional subspaces, products, etc. A barrel in a locally convex space is a closed * We are grateful for support and hospitality from EPSRC GRÂL67257 and Strathclyde University, and for helpful discussions with Professors J. -
Quasi-Barrelled Locally Convex Spaces 811
i960] quasi-barrelled locally convex spaces 811 Bibliography 1. R. E. Lane, Absolute convergence of continued fractions, Proc. Amer. Math. Soc. vol. 3 (1952) pp. 904-913. 2. R. E. Lane and H. S. Wall, Continued fractions with absolutely convergent even and odd parts, Trans. Amer. Math. Soc. vol. 67 (1949) pp. 368-380. 3. W. T. Scott and H. S. Wall, A convergence theorem for continued fractions, Trans. Amer. Math. Soc. vol. 47 (1940) pp. 155-172. 4. H. S. Wall, Analytic theory of continued fractions, New York, D. Van Nostrand Company, Inc., 1948. The University of Texas and University of Houston QUASI-BARRELLED LOCALLY CONVEX SPACES MARK MAHOWALD AND GERALD GOULD 1. Introduction and preliminary definitions. The main object of this paper is to answer some problems posed by Dieudonné in his paper Denumerability conditions in locally convex vector spaces [l]. His two main results are as follows: Proposition 1. If Eis a barrelled space on which there is a countable fundamental system of convex compact subsets, [Definition 1.2] then it is the strong dual of a Fréchet-Montel Space. Proposition 2. If E is either bornological or barrelled, and if there is a countable fundamental system of compact subsets, then E is dense in the strong dual of a Fréchet-Montel Space. Two questions raised by Dieudonné in connection with these results are: (a) If E is either bornological or barrelled then it is certainly quasi- barrelled [l, Chapter 3, §2, Example 12]. Can one substitute this weaker condition on E in Proposition 2? (b) Is there is an example of a quasi-barrelled space which is neither barrelled nor bornological? We shall show that the answer to (a) is "Yes," and that the answer to (b) is also "Yes," so that the generalization is in fact a real one. -
Canonical Metric on Moduli Spaces of Log Calabi-Yau Varieties Hassan Jolany
Canonical metric on moduli spaces of log Calabi-Yau varieties Hassan Jolany To cite this version: Hassan Jolany. Canonical metric on moduli spaces of log Calabi-Yau varieties . 2017. hal-01413746v3 HAL Id: hal-01413746 https://hal.archives-ouvertes.fr/hal-01413746v3 Preprint submitted on 14 Feb 2017 (v3), last revised 20 Jun 2017 (v4) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Canonical metric on moduli spaces of log Calabi-Yau varieties Hassan Jolany February 14, 2017 Abstract In this paper, we give a short proof of closed formula [9],[18] of loga- rithmic Weil-Petersson metric on moduli space of log Calabi-Yau varieties (if exists!) of conic and Poincare singularities and its connection with Bismut-Vergne localization formula. Moreover we give a relation between logarithmic Weil-Petersson metric and the logarithmic version of semi Ricci flat metric on the family of log Calabi-Yau pairs with conical sin- gularities. In final we consider the semi-positivity of singular logarithmic Weil-Petersson metric on the moduli space of log-Calabi-Yau varieties. Moreover, we show that Song-Tian-Tsuji measure is bounded along Iitaka fibration if and only if central fiber has log terminal singularities and we consider the goodness of fiberwise Calabi-Yau metric in the sense of Mumford and goodness of singular Hermitian metric corresponding to Song-Tian-Tsuji measure. -
Topological Vector Spaces
An introduction to some aspects of functional analysis, 3: Topological vector spaces Stephen Semmes Rice University Abstract In these notes, we give an overview of some aspects of topological vector spaces, including the use of nets and filters. Contents 1 Basic notions 3 2 Translations and dilations 4 3 Separation conditions 4 4 Bounded sets 6 5 Norms 7 6 Lp Spaces 8 7 Balanced sets 10 8 The absorbing property 11 9 Seminorms 11 10 An example 13 11 Local convexity 13 12 Metrizability 14 13 Continuous linear functionals 16 14 The Hahn–Banach theorem 17 15 Weak topologies 18 1 16 The weak∗ topology 19 17 Weak∗ compactness 21 18 Uniform boundedness 22 19 Totally bounded sets 23 20 Another example 25 21 Variations 27 22 Continuous functions 28 23 Nets 29 24 Cauchy nets 30 25 Completeness 32 26 Filters 34 27 Cauchy filters 35 28 Compactness 37 29 Ultrafilters 38 30 A technical point 40 31 Dual spaces 42 32 Bounded linear mappings 43 33 Bounded linear functionals 44 34 The dual norm 45 35 The second dual 46 36 Bounded sequences 47 37 Continuous extensions 48 38 Sublinear functions 49 39 Hahn–Banach, revisited 50 40 Convex cones 51 2 References 52 1 Basic notions Let V be a vector space over the real or complex numbers, and suppose that V is also equipped with a topological structure. In order for V to be a topological vector space, we ask that the topological and vector spaces structures on V be compatible with each other, in the sense that the vector space operations be continuous mappings.