Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic
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Arxiv:Math/9201256V1 [Math.RT] 1 Jan 1992 Naso Lblaayi N Geometry and Vol
Annals of Global Analysis and Geometry Vol. 8, No. 3 (1990), 299–313 THE MOMENT MAPPING FOR UNITARY REPRESENTATIONS Peter W. Michor Institut f¨ur Mathematik der Universit¨at Wien, Austria Abstract. For any unitary representation of an arbitrary Lie group I construct a moment mapping from the space of smooth vectors of the representation into the dual of the Lie algebra. This moment mapping is equivariant and smooth. For the space of analytic vectors the same construction is possible and leads to a real analytic moment mapping. Table of contents 1.Introduction . .. .. .. .. .. .. .. .. .. .. 1 2.Calculusofsmoothmappings . 2 3.Calculusofholomorphicmappings . 5 4.Calculusofrealanalyticmappings . 6 5.TheSpaceofSmoothVectors . 7 6.Themodelforthemomentmapping . 8 7. Hamiltonian Mechanics on H∞ .................... 9 8. The moment mapping for a unitary representation . .... 11 9.Therealanalyticmomentmapping . 13 1. Introduction arXiv:math/9201256v1 [math.RT] 1 Jan 1992 With the help of the cartesian closed calculus for smooth mappings as explained in [F-K] we can show, that for any Lie group and for any unitary representation its restriction to the space of smooth vectors is smooth. The imaginary part of the hermitian inner product restricts to a ”weak” symplectic structure on the vector space of smooth vectors. This gives rise to the Poisson bracket on a suitably chosen space of smooth functions on the space of smooth vectors. The derivative of the representation on the space of smooth vectors is a symplectic action of the Lie algebra, which can be lifted to a Hamiltonian action, i.e. a Lie algebra homomorphism from the Lie algebra into the function space with the Poisson bracket. -
Uniqueness of Von Neumann Bornology in Locally C∗-Algebras
Scientiae Mathematicae Japonicae Online, e-2009 91 UNIQUENESS OF VON NEUMANN BORNOLOGY IN LOCALLY C∗-ALGEBRAS. A BORNOLOGICAL ANALOGUE OF JOHNSON’S THEOREM M. Oudadess Received May 31, 2008; revised March 20, 2009 Abstract. All locally C∗- structures on a commutative complex algebra have the same bound structure. It is also shown that a Mackey complete C∗-convex algebra is semisimple. By the well-known Johnson’s theorem [4], there is on a given complex semi-simple algebra a unique (up to an isomorphism) Banach algebra norm. R. C. Carpenter extended this result to commutative Fr´echet locally m-convex algebras [3]. Without metrizability, it is not any more valid even in the rich context of locally C∗-convex algebras. Below there are given telling examples where even a C∗-algebra structure is involved. We follow the terminology of [5], pp. 101-102. Let E be an involutive algebra and p a vector space seminorm on E. We say that p is a C∗-seminorm if p(x∗x)=[p(x)]2, for every x. An involutive topological algebra whose topology is defined by a (saturated) family of C∗-seminorms is called a C∗-convex algebra. A complete C∗-convex algebra is called a locally C∗-algebra (by Inoue). A Fr´echet C∗-convex algebra is a metrizable C∗- convex algebra, that is equivalently a metrizable locally C∗-algebra, or also a Fr´echet locally C∗-algebra. All the bornological notions can be found in [6]. The references for m-convexity are [5], [8] and [9]. Let us recall for convenience that the bounded structure (bornology) of a locally convex algebra (l.c.a.)(E,τ) is the collection Bτ of all the subsets B of E which are bounded in the sense of Kolmogorov and von Neumann, that is B is absorbed by every neighborhood of the origin (see e.g. -
Infinite Dimensional Lie Theory from the Point of View of Functional
Infinite dimensional Lie Theory from the point of view of Functional Analysis Josef Teichmann F¨urmeine Eltern, Michael, Michaela und Hannah Zsofia Hazel. Abstract. Convenient analysis is enlarged by a powerful theory of Hille-Yosida type. More precisely asymptotic spectral properties of bounded operators on a convenient vector space are related to the existence of smooth semigroups in a necessary and sufficient way. An approximation theorem of Trotter-type is proved, too. This approximation theorem is in fact an existence theorem for smooth right evolutions of non-autonomous differential equations on convenient locally convex spaces and crucial for the following applications. To enlighten the generically "unsolved" (even though H. Omori et al. gave interesting and concise conditions for regularity) question of the existence of product integrals on convenient Lie groups, we provide by the given approximation formula some simple criteria. On the one hand linearization is used, on the other hand remarkable families of right invariant distance functions, which exist on all up to now known Lie groups, are the ingredients: Assuming some natural global conditions regularity can be proved on convenient Lie groups. The existence of product integrals is an essential basis for Lie theory in the convenient setting, since generically differential equations cannot be solved on non-normable locally convex spaces. The relationship between infinite dimensional Lie algebras and Lie groups, which is well under- stood in the regular case, is also reviewed from the point of view of local Lie groups: Namely the question under which conditions the existence of a local Lie group for a given convenient Lie algebra implies the existence of a global Lie group is treated by cohomological methods. -
On Bornivorous Set
On Bornivorous Set By Fatima Kamil Majeed Al-Basri University of Al-Qadisiyah College Of Education Department of Mathematics E-mail:[email protected] Abstract :In this paper, we introduce the concept of the bornivorous set and its properties to construct bornological topological space .Also, we introduce and study the properties related to this concepts like bornological base, bornological subbase , bornological closure set, bornological interior set, bornological frontier set and bornological subspace . Key words : bornivorous set , bornological topological space,b-open set 1.Introduction- The space of entire functions over the complex field C was introduced by Patwardhan who defined a metric on this space by introducing a real-valued map on it[6]. In(1971), H.Hogbe- Nlend introduced the concepts of bornology on a set [3].Many workers such as Dierolf and Domanski, Jan Haluska and others had studied various bornological properties[2]. In this paper at the second section ,bornivorous set has been introduced with some related concepts. While in the third section a new space “Bornological topological space“ has been defined and created in the base of bornivorous set . The bornological topological space also has been explored and its properties .The study also extended to the concepts of the bornological base and bornological subbase of bornological topological space .In the last section a new concepts like bornological closure set, bornological drived set, bornological dense set, bornological interior set, bornological exterior set, bornological frontier set and bornological topological subspace, have been studied with supplementary properties and results which related to them. 1 Definition1.1[3] Let A and B be two subsets of a vector space E. -
Lecture Notes 1
Convenient Calculus and Differential Geometry in Infinite Dimensions Peter W. Michor Norwegian Summer School on Analysis and Geometry Bergen, Norway. June 24-28, 2013 Some words on smooth convenient calculus Traditional differential calculus works well for finite dimensional vector spaces and for Banach spaces. Beyond Banach spaces, the main difficulty is that composition of linear mappings stops to be jointly continuous at the level of Banach spaces, for any compatible topology. For more general locally convex spaces we sketch here the convenient approach as explained in [Fr¨olicher-Kriegl1988] and [Kriegl-Michor 1997]. The c1-topology Let E be a locally convex vector space. A curve c : R ! E is called smooth or C 1 if all derivatives exist and are continuous. Let 1 C (R; E) be the space of smooth functions. It can be shown that 1 the set C (R; E) does not depend on the locally convex topology of E, only on its associated bornology (system of bounded sets). The final topologies with respect to the following sets of mappings into E coincide: 1 1. C (R; E). 2. The set of all Lipschitz curves (so that c(t)−c(s) f t−s : t 6= s; jtj; jsj ≤ Cg is bounded in E, for each C). 3. The set of injections EB ! E where B runs through all bounded absolutely convex subsets in E, and where EB is the linear span of B equipped with the Minkowski functional kxkB := inffλ > 0 : x 2 λBg. 4. The set of all Mackey-convergent sequences xn ! x (there exists a sequence 0 < λn % 1 with λn(xn − x) bounded). -
Closed Graph Theorems for Bornological Spaces
Khayyam J. Math. 2 (2016), no. 1, 81{111 CLOSED GRAPH THEOREMS FOR BORNOLOGICAL SPACES FEDERICO BAMBOZZI1 Communicated by A. Peralta Abstract. The aim of this paper is that of discussing closed graph theorems for bornological vector spaces in a self-contained way, hoping to make the subject more accessible to non-experts. We will see how to easily adapt classical arguments of functional analysis over R and C to deduce closed graph theorems for bornological vector spaces over any complete, non-trivially valued field, hence encompassing the non-Archimedean case too. We will end this survey by discussing some applications. In particular, we will prove De Wilde's Theorem for non-Archimedean locally convex spaces and then deduce some results about the automatic boundedness of algebra morphisms for a class of bornological algebras of interest in analytic geometry, both Archimedean (complex analytic geometry) and non-Archimedean. Introduction This paper aims to discuss the closed graph theorems for bornological vector spaces in a self-contained exposition and to fill a gap in the literature about the non-Archimedean side of the theory at the same time. In functional analysis over R or C bornological vector spaces have been used since a long time ago, without becoming a mainstream tool. It is probably for this reason that bornological vector spaces over non-Archimedean valued fields were rarely considered. Over the last years, for several reasons, bornological vector spaces have drawn new attentions: see for example [1], [2], [3], [5], [15] and [22]. These new developments involve the non-Archimedean side of the theory too and that is why it needs adequate foundations. -
Notes, We Review Useful Concepts in Convexity and Normed Spaces
Mastermath, Spring 2019 Lecturers: D. Dadush, J. Briet Geometric Functional Analysis Lecture 0 Scribe: D. Dadush Review of Convexity and Normed Spaces In these notes, we review useful concepts in convexity and normed spaces. As a warning, many of the statements made here will not be proved. For detailed proofs, one is recommended to look up reference books in functional and convex analysis. The intent here is for these notes to serve as a reference for the basic language for the course. The reader may thus skim the material upon first reading, and then reread more carefully later during the course if the relevant concepts remain unclear. 1 Basic Convexity In this section, we will focus on convexity theorems in a finite dimensional real vector space, which will identify with Rn or subspaces theoreof. For vectors x, y 2 Rn, we define the standard T n n inner product hx, yi := x y = ∑i=1 xiyi. For two sets A, B ⊆ R and scalars c, d 2 R, we define the Minkowski sum cA + dB = fca + db : a 2 A, b 2 Bg. A convex set K ⊆ Rn is a set for which the line segment between any two points is in the set, or formally for which 8x, y 2 K, l 2 [0, 1], lx + (1 − l)y 2 K. A useful equivalent definition is that a non-empty set K is convex if 8a, b ≥ 0, aK + bK = (a + b)K. If K = −K, K is said to be symmetric. If K is compact with non-empty interior, then K is a convex body. -
Bornological Topology Space Separation Axioms a Research Submitted by Deyar
Republic of Iraq Ministry of Higher Education & Scientific Research AL-Qadisiyah University College of Computer Science and Mathematics Department of Mathematics Bornological Topology Space Separation Axioms A Research Submitted by Deyar To the Council of the department of Mathematics ∕ College of Education, University of AL-Qadisiyah as a Partial Fulfilment of the Requirements for the Bachelor Degree in Mathematics Supervised by Fatma Kamel Majeed A. D. 2019 A.H. 1440 Abstract we study Bornological Topology Separation Axioms like bornological topology , bornological topology , bornological topology , bornological topology , bornological topology and the main propositions and theorems about this concept. introduction For the first time in (1977), H. Hogbe–NIend [1] introduced the Concept of Bornology on a set and study Bornological Construction. In chapter one study Bornology on a set , Bornological subspace, convex Bornological space, Bornological vector space and Bornivorous set. Bornological topology space were first introduced and investigated in [4], we introduce in chapter two Bornological topology space and we study Bornological topology continuous and bornological topology homeomorphism. Bornological topology open map, bornological topology separation axioms studied in chapter three like bornological topology , bornological topology , bornological topology And bornological topology Bornological topology and main properties have been studied. The Contents Subject Page Chapter One 1.1 Bornological Space 1 1. 2 Bornivorous Set 4 Chapter Two 2.1 Bornological Topological Space 6 2.2 Bornological Topology Continuous 8 Chapter three 3.1 Bornological topology And Bornological 9 topology 3.2 Bornological topology , Bornological topology 10 And Bornological topology Chapter One 1.1 Bornological space In this section, we introduce some definitions, bornological space, bornological vector space, convex bornological vector space, separated bornological vector space, bounded map and some examples . -
The Convenient Setting for Denjoy--Carleman Differentiable
THE CONVENIENT SETTING FOR DENJOY–CARLEMAN DIFFERENTIABLE MAPPINGS OF BEURLING AND ROUMIEU TYPE ANDREAS KRIEGL, PETER W. MICHOR, AND ARMIN RAINER Abstract. We prove in a uniform way that all Denjoy–Carleman differen- tiable function classes of Beurling type C(M) and of Roumieu type C{M}, admit a convenient setting if the weight sequence M = (Mk) is log-convex and of moderate growth: For C denoting either C(M) or C{M}, the category of C-mappings is cartesian closed in the sense that C(E, C(F,G)) =∼ C(E × F,G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of C-diffeomorphisms is a regular C-Lie group if C ⊇ Cω, but not better. 1. Introduction Denjoy–Carleman differentiable functions form classes of smooth functions that are described by growth conditions on the Taylor expansion. The growth is pre- scribed in terms of a sequence M = (Mk) of positive real numbers which serves as a weight for the iterated derivatives: for compact K the sets f (k)(x) k : x K, k N ρ k! Mk ∈ ∈ n o are required to be bounded. The positive real number ρ is subject to either a universal or an existential quantifier, thereby dividing the Denjoy–Carleman classes into those of Beurling type, denoted by C(M), and those of Roumieu type, denoted {M} by C , respectively. For the constant sequence M = (Mk) = (1), as Beurling type we recover the real and imaginary parts of all entire functions on the one hand, and as Roumieu type the real analytic functions on the other hand, where 1/ρ plays the role of a radius of convergence. -
A Homological Study of Bornological Spaces
Pr´epublications Math´ematiques de l'Universit´e Paris 13 A Homological Study of Bornological Spaces by Fabienne Prosmans Jean-Pierre Schneiders 00-21 December 2000 Laboratoire Analyse, G´eom´etrie et Applications, UMR 7539 Institut Galil´ee, Universit´e Paris 13 93430 Villetaneuse (France) A Homological Study of Bornological Spaces Fabienne Prosmans Jean-Pierre Schneiders December 15, 2000 Abstract In this paper, we show that the category Bc of bornological vector spaces _ of convex type and the full subcategories B c and Bc formed by separated and complete objects form quasi-abelian categories. This allows us to study them from a homological point of view. In particular,bwe characterize acyclic _ inductive systems and prove that although the categories Bc (resp. B c, Ban) and the categories Ind(Sns) (resp. Ind(N vs), Ind(Ban)) formed by the ind- objects of the category of semi-normed vector spaces (resp. normed vector spaces, Banach spaces) are not equivalent, there is an equivalence between their derived categories given by a canonical triangulated functor which pre- serves the left t-structures. In particular these categories have the same left heart; a fact which means roughly that they have the same homological alge- bra. As a consequence, we get a link between the theory of sheaves of complete bornological spaces and that of sheaves with values in Ind(Ban) used in [4]. 0 Introduction In this paper, we study from the point of view of homological algebra the category _ Bc of bornological vector spaces of convex type and its full subcategories B c and Bc formed by separated and complete objects. -
Approximating Euler's Equation on the Full Diffeomorphism Group, And
Approximating Euler's equation on the full diffeomorphism group, and soliton-like solutions Peter W. Michor Geometrical Aspects of Hydrodynamics workshop Simons Center for Geometry and Physics Stony Brook University USA Based on [David Mumford, Peter W. Michor: On Euler's equation and `EPDiff'. Journal of Geometric Mechanics, 5, 3 (2013), 319-344.] May 19 - 23, 2014 Some words on smooth convenient calculus Traditional differential calculus works well for finite dimensional vector spaces and for Banach spaces. Beyond Banach spaces, the main difficulty is that composition of linear mappings stops to be jointly continuous at the level of Banach spaces, for any compatible topology. For more general locally convex spaces we sketch here the convenient approach as explained in [Fr¨olicher-Kriegl1988] and [Kriegl-Michor 1997]. The c1-topology Let E be a locally convex vector space. A curve c : R ! E is called smooth or C 1 if all derivatives exist and are continuous. Let 1 C (R; E) be the space of smooth functions. It can be shown that 1 the set C (R; E) does not depend entirely on the locally convex topology of E, only on its associated bornology (system of bounded sets). The final topologies with respect to the following sets of mappings into E coincide: 1 1. C (R; E). 2. The set of all Lipschitz curves (so that c(t)−c(s) f t−s : t 6= s; jtj; jsj ≤ Cg is bounded in E, for each C). 3. The set of injections EB ! E where B runs through all bounded absolutely convex subsets in E, and where EB is the linear span of B equipped with the Minkowski functional kxkB := inffλ > 0 : x 2 λBg. -
A Zoo of Diffeomorphism Groups on $$ \Mathbb{R }^{N}$$ Rn
A zoo of diffeomorphism groups on $$ \mathbb{R }^{n}$$ Rn Peter W. Michor & David Mumford Annals of Global Analysis and Geometry ISSN 0232-704X Ann Glob Anal Geom DOI 10.1007/s10455-013-9380-2 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media Dordrecht. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Ann Glob Anal Geom DOI 10.1007/s10455-013-9380-2 A zoo of diffeomorphism groups on Rn Peter W. Michor · David Mumford Received: 25 November 2012 / Accepted: 6 May 2013 © Springer Science+Business Media Dordrecht 2013 n n n Abstract We consider the groups DiffB(R ), Diff H ∞ (R ), and DiffS (R ) of smooth diffeo- n morphisms on R which differ from the identity by a function which is in either B (bounded ∞ = k S in all derivatives), H k≥0 H ,or (rapidly decreasing). We show that all these groups are smooth regular Lie groups.