Holomorphic Functions of Exponential Type and Duality for Stein Groups
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International Centre for Theoretical Physics
zrrl^^^'C^z ic/89/30i INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ON THE APPROXIMATIVE NORMAL VALUES OF MULTIVALUES OPERATORS IN TOPOLOGICAL VECTOR SPACE Nguyen Minh Chuong and INTERNATIONAL ATOMIC ENERGY Khuat van Ninh AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/89/301 SUNTO International Atomic Energy Agency and In questo articolo si considera il problemadell'approssimaztone di valori normali di oper- United Nations Educational Scientific and Cultural Organization atori chiusi lineari multivoci dallo spazio vettoriale topologico de Mackey, a valori in un E-spazio. INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS Si dimostrano l'esistenza di un valore normale e la convergenza dei valori approssimanti il valore normale. ON THE APPROXIMATIVE NORMAL VALUES OF MULTIVALUED OPERATORS IN TOPOLOGICAL VECTOR SPACE * Nguyen Minn Chuong * International Centre for Theoretical Physics, Trieste, Italy. and Khuat van Ninh Department of Numerical Analysis, Institute of Mathematics, P.O. Box 631, Bo Ho, 10 000 Hanoi, Vietnam. ABSTRACT In this paper the problem of approximation of normal values of multivalued linear closed operators from topological vector Mackey space into E-space is considered. Existence of normal value and covergence of approximative values to normal value are proved. MIRAMARE - TRIESTE September 1989 To be submitted for publication. Permanent address: Department of Numerical Analysis, Institute of Mathematics, P.O. Box 631, Bo Ho 10 000 Hanoi, Vietnam. 1. Introduction Let X , Y be closed Eubspaces of X,Y respectively, where X is normed In recent years problems on normal value and on approximation of space and Y is Hackey space. Let T be a linear multivalued operator from normal value of multivalued linear operators have been investigated Y to X . -
Lectures on Analytic Geometry Peter Scholze (All Results Joint with Dustin
Lectures on Analytic Geometry Peter Scholze (all results joint with Dustin Clausen) Contents Analytic Geometry 5 Preface 5 1. Lecture I: Introduction 6 2. Lecture II: Solid modules 11 3. Lecture III: Condensed R-vector spaces 16 4. Lecture IV: M-complete condensed R-vector spaces 20 Appendix to Lecture IV: Quasiseparated condensed sets 26 + 5. Lecture V: Entropy and a real BdR 28 6. Lecture VI: Statement of main result 33 Appendix to Lecture VI: Recollections on analytic rings 39 7. Lecture VII: Z((T ))>r is a principal ideal domain 42 8. Lecture VIII: Reduction to \Banach spaces" 47 Appendix to Lecture VIII: Completions of normed abelian groups 54 Appendix to Lecture VIII: Derived inverse limits 56 9. Lecture IX: End of proof 57 Appendix to Lecture IX: Some normed homological algebra 65 10. Lecture X: Some computations with liquid modules 69 11. Lecture XI: Towards localization 73 12. Lecture XII: Localizations 79 Appendix to Lecture XII: Topological invariance of analytic ring structures 86 Appendix to Lecture XII: Frobenius 89 Appendix to Lecture XII: Normalizations of analytic animated rings 93 13. Lecture XIII: Analytic spaces 95 14. Lecture XIV: Varia 103 Bibliography 109 3 Analytic Geometry Preface These are lectures notes for a course on analytic geometry taught in the winter term 2019/20 at the University of Bonn. The material presented is part of joint work with Dustin Clausen. The goal of this course is to use the formalism of analytic rings as defined in the course on condensed mathematics to define a category of analytic spaces that contains (for example) adic spaces and complex-analytic spaces, and to adapt the basics of algebraic geometry to this context; in particular, the theory of quasicoherent sheaves. -
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. -
181. on Nuclear Spaces with Fundamental System O F Bounded Sets
No. 8] Proc. Japan Acad., 44 (1968) 807 181. On Nuclear Spaces with Fundamental System o f Bounded Sets. II By Shunsuke FUNAKOs1 (Comm. by Kinjiro KuNUGI, M. J. A., Oct. 12, 1968) A locally convex vector space with a countable fundamental system of bounded sets has already been developed in several bibliog- raphies. Barrelled spaces and quasi-barrelled spaces with a count- able fundamental system of compact sets has been studied by J. Dieudonne [2] and by M. Mahowald and G. Gould [7] respectively. We considered, the open mapring and closed graph theorems on a nuclear dualmetric space in the previous paper [4]. Let E be a nonmed space then E is a nuclear space if and only if it is finite dimentional. It is also known that a nonmed space can only be a Montel (i.e., barrelled and perfect) space if it is finite dimen- sional. In this paper, we prove a nuclear dualmetric space which is quasi-complete is Montel space, and using this result, we consider analogous theorem to M. Mahowald and G. Gould [7], in nuclear space. For nuclear spaces and its related notion, see A. Pietsch [8] and S. Funakosi [4]. Most of the definitions and notations of the locally convex vector spaces are taken' from N. Bourbaki [1] and T. Husain [5]. Definition. Let E be a locally convex space and E' its dual. (1) I f only all countable strong bounded subset o f E' are equi- continuous, then E is called the oW-quasi-barrelled. (2) Let E be a a-quasi-barrelled space, if there exists a countable fundamental system o f bounded subset in E, then E is called the dual- metric space. -
If(Z)
COMPATIBLE SEMINORMS IN A VECTOR LATTICE* BY CASPER GOFFMAN UNIVERSITY OF OKLAHOMA Communicated by S. Bochner, May 16, 1956 Let X be a vector space. A locally convex topology r in X is given by a system of seminorms pal A, for which XJ is a Hausdorff space. We suppose that there is an order relation w in X for which X<, is a vector lattice. We suppose, further, that the order dual, X,', which consists of all bounded linear forms on Xs, is a total subspace of the algebraic dual X' of X, i.e., for every x $ 0 there is a bounded f such that f(x) $ 0. In particular, this implies that X, is Archimedean. A seminorm p is compatible with w if x| > yI implies p(x) > p(y). A locally convex topology r is compatible with co if its seminorms may be chosen so as to be compatible with W. For general information see N. Bourbaki, Elements de mathcmatique.' Under the above assumptions, we observe that for every w there is a finest r compatible with w. In particular, we show that there is at most one Banach topology compatible with w, and, if it exists, it is the finest locally convex topology compatible with w. Thus, for various function spaces, the norm in use is seen to yield the only possible Banach space compatible with the natural order relation of the space. We first note that if p is a seminorm compatible with w, then the continuity semi- norm set of p is contained in X,'. -
Surjections in Locally Convex Spaces
Ghent University Faculty of Sciences Department of Mathematics Surjections in locally convex spaces by Lenny Neyt Supervisor: Prof. Dr. H. Vernaeve Master dissertation submitted to the Faculty of Sciences to obtain the academic degree of Master of Science in Mathematics: Pure Mathematics. Academic Year 2015{2016 Preface For me, in the past five years, mathematics metamorphosed from an interest into my greatest passion. I find it truly fascinating how the combination of logical thinking and creativity can produce such rich theories. Hopefully, this master thesis conveys this sentiment. I hereby wish to thank my promotor Hans Vernaeve for the support and advice he gave me, not only during this thesis, but over the last three years. Furthermore, I thank Jasson Vindas and Andreas Debrouwere for their help and suggestions. Last but not least, I wish to express my gratitude toward my parents for the support and stimulation they gave me throughout my entire education. The author gives his permission to make this work available for consultation and to copy parts of the work for personal use. Any other use is bound by the restrictions of copyright legislation, in particular regarding the obligation to specify the source when using the results of this work. Ghent, May 31, 2016 Lenny Neyt i Contents Preface i Introduction 1 1 Preliminaries 3 1.1 Topology . 3 1.2 Topological vector spaces . 7 1.3 Banach and Fr´echet spaces . 11 1.4 The Hahn-Banach theorem . 13 1.5 Baire categories . 15 1.6 Schauder's theorem . 16 2 Locally convex spaces 18 2.1 Definition . -
The Mackey Topology As a Mixed Topology 109
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 53, Number I, November 1975 THE MACKEYTOPOLOGY AS A MIXEDTOPOLOGY J. B. COOPER ABSTRACT. A theorem on the coincidence of a mixed topology and the Mackey topology is given. The techniques of partitions of unity are used. As a corollary, a result of LeCam and Conway on the strict topology is obtained. Similar methods are applied to obtain a theorem of Collins and Dorroh. Introduction. In recent years the space C(S) of bounded, continuous complex-valued functions on a locally compact space S with the strict topol- ogy has received a great deal of attention. In [5] we have shown that this to- pology was a special case of a mixed topology in the sense of the Polish school. Many of the results on the strict topology are, in fact, special cases of general results on mixed topologies and, when this is the case, it seems to us to be desirable to present this approach since it displays the strict to- pology as part of a larger scheme rather than as an isolated phenomenon. In this note, we give an example of this process by considering the prob- lem: When is the strict topology on C(S) the Mackey topology? LeCam and Conway have shown that this is the case when S is paracompact. In fact, there are a number of results of this type, that is, identifying mixed topologies with the Mackey topology (see Cooper [6], Stroyan Ll3]) and this suggests that there might be a general theorem of this type in the context of mixed to- pologies. -
The Strict Topology and Compactness in the Space of Measures
Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1965 The trS ict Topology and Compactness in the Space of Measures. John Bligh Conway Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Conway, John Bligh, "The trS ict Topology and Compactness in the Space of Measures." (1965). LSU Historical Dissertations and Theses. 1068. https://digitalcommons.lsu.edu/gradschool_disstheses/1068 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. This dissertation has been microfilmed exactly as received 6 6—7 24 CONWAY, John Bligh, 1939- THE STRICT TOPOLOGY AND COMPACTNESS IN THE SPACE OF MEASURES. Louisiana State University, Ph.D., 1965 Mathematics University Microfilms, Inc., Ann Arbor, Michigan Reproduced with permission of the copyright owner. Further reproduction prohibitedpermission. without THE STRICT TOPOLOGY AND COMPACTNESS IN THE SPACE OF MEASURES A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics John B/^Conway B.S., Loyola University, 196I August, 1965 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENT The author wishes to express his appreciation to Professor Heron S. Collins for his advice and encourage ment. This dissertation was written while the author held a National Science Foundation Cooperative Fellowship. -
Mathematics in Computation •
SOCIETY THE AMERICAN MATHEMATICAL SOCIETY Edited by John W. Green and Gordon L. Walker CONTENTS MEETINGS Calendar of Meetings . • . .. • . • . 64 Program of the April Meeting in Chicago, Illinois . • . 65 Abstracts of the Meeting - pages 104- 117 Program of the April Meeting in Atlantic City, New Jersey ......•..... 71 Abstracts of the Meeting - pages 118-130 Program of the April Meeting in Monterey, California. • . • . 78 Abstracts of the Meeting- pages 131-138 PRELIMINARY ANNOUNCEMENT OF MEETING .....•....... 82 A REPORT TO MEMBERS ........•.....•............. 85 NEWS ITEMS AND ANNOUNCEMENTS .•.....•.........•.•.... 81, 88, 9Z, 94 PERSONAL ITEMS . • . • . • . • . • . • . 89 NEW AMS PUBLICATIONS . • . • . 93 LETTERS TO THE EDITOR ........•. 95 MEMORANDA TO MEMBERS Mathematics in Computation • . • . 96 Reprinting of Back Volumes of Mathematical Reviews . • . 96 Reciprocity Agreement With the Sociedade de Matematica de Sao Paulo . 96 Change of Address Notification Deadlines for Journals ....• , . • . 96 Group Travel to Stockholm . • . • . • . • . • . • . 97 CATALOG OF LECTURE NOTES . • . • . • • . • . 99 SUPPLEMENTARY PROGRAM -No. 10. 100 ABSTRACTS OF CONTRIBUTED PAPERS ...........•••............. 103 ERRATA - Volume 9. .. • . 155 INDEX TO ADVERTISERS. • . • . 163 RESERVATION FORM . • . • . • . 163 MEETINGS CALENDAR OF MEETINGS Note: This Calendar lists all of the meetings which have been approved by the Council up to the date at which this issue of the NOTICES was sent to press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the American Mathematical Society. The meeting dates which fall rather far in the future are subject to change. This is particularly true of the meetings to which no numbers have yet been assigned. Meet Deadline ing Date Place for No. Abstracts* (june Issue of the NOTICES) April 27 592 August 27-31, 1962 Vancouver, British Columbia july 6 (67th Summer Meeting) 593 October Z7, 1962 Hanover, New Hampshire Sept. -
ON SEMI-BARRELLED SPACES Henri Bourlès
ON SEMI-BARRELLED SPACES Henri Bourlès To cite this version: Henri Bourlès. ON SEMI-BARRELLED SPACES. 2013. hal-02368293 HAL Id: hal-02368293 https://hal.archives-ouvertes.fr/hal-02368293 Submitted on 18 Nov 2019 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. ON SEMI-BARRELLED SPACES HENRI BOURLES` Abstract. The aim of this paper is to clarify the properties of semi- barrelled spaces (also called countably quasi-barrelled spaces in the lit- erature). These spaces were studied by several authors, in particular in the classical book of N. Bourbaki ”Espaces vectoriels topologiques”. However, six incorrect statements can be found in this reference. In par- ticular: a Hausdorff and quasi-complete semi-barrelled space is complete, a semi-barrelled, semi-reflexive space is complete, a locally convex hull of semi-barrelled semi-reflexive spaces is semi-reflexive, a locally con- vex hull of semi-barrelled reflexive spaces is reflexive. We show through counterexamples that these statements are false. To conclude, we show how these false claims can be corrected and we collect some properties of semi-barrelled spaces. 1. Introduction Semi-barrelled spaces are studied in the classical book of N. -
Emerging Applications of Geometric Multiscale Analysis 211
ICM 2002 Vol. I 209–233 · · Emerging Applications of Geometric Multiscale Analysis David L. Donoho∗ Abstract Classical multiscale analysis based on wavelets has a number of successful applications, e.g. in data compression, fast algorithms, and noise removal. Wavelets, however, are adapted to point singularities, and many phenom- ena in several variables exhibit intermediate-dimensional singularities, such as edges, filaments, and sheets. This suggests that in higher dimensions, wavelets ought to be replaced in certain applications by multiscale analysis adapted to intermediate-dimensional singularities, My lecture described various initial attempts in this direction. In partic- ular, I discussed two approaches to geometric multiscale analysis originally arising in the work of Harmonic Analysts Hart Smith and Peter Jones (and others): (a) a directional wavelet transform based on parabolic dilations; and (b) analysis via anistropic strips. Perhaps surprisingly, these tools have po- tential applications in data compression, inverse problems, noise removal, and signal detection; applied mathematicians, statisticians, and engineers are ea- gerly pursuing these leads. Note: Owing to space constraints, the article is a severely compressed version of the talk. An extended version of this article, with figures used in the presentation, is available online at: http://www-stat.stanford.edu/∼donoho/Lectures/ICM2002 2000 Mathematics Subject Classification: 41A30, 41A58, 41A63, 62G07, 62G08, 94A08, 94A11, 94A12, 94A29. Keywords and Phrases: Harmonic analysis, Multiscale analysis, Wavelets, Ridgelets, Curvelets, Directional wavelets. arXiv:math/0212395v1 [math.ST] 1 Dec 2002 1. Prologue Since the last ICM, we have lost three great mathematical scientists of the twentieth century: Alberto Pedro Calder´on (1922-1999), John Wilder Tukey (1915- 2000) and Claude Elwood Shannon (1916-2001).