DOCSLIB.ORG

Mixed Limits of Some Functional Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mixed Limits of Some Functional Spaces Mixed limits of some functional spaces A.A. Novikov∗, Z.Eskandarian◦, Z.Kholmatova† February 26, 2021 ∗ e-mail: [email protected], Kazan Federal University, Kremlievskaia ul. 18, Kazan, Tatarstan, 420008, Russia ◦ e-mail: [email protected],Kazan Federal University, Kremlievskaia ul. 18, Kazan, Tatarstan, 420008, Russia † e-mail: [email protected], Innopolis University, Universitetskaia ul., 1, Innopolis, Tatarstan, 420500, Russia Abstract In this article we propose a conception of mixed limits of functional spaces as the case, when the upper limit (projective limit of inductive limits) and the lower limit (inductive limit of projective limits) coincide as topological spaces, which are generalization of inductive and projective limits of functional spaces. We show a cases where these mixed limits are naturally obtained as the limit spaces of non-commutative Lp-type spaces associated with the sequence of operators. Also, we obtain results on the properties of limit spaces, we show that limit spaces of Banach algebras are (LF)-spaces, if they converge. subject classification: 46A13, 46B70, 46L10, 46L51, 46L52 keywords: inductive limit, projective limit, power parameter, (LB)-space, (LF)-space, Frechet space, locally convex space, order unit base norm, inductive arXiv:1901.09276v3 [math.OA] 25 Feb 2021 limit, initial topology, final topology, order unit space, measurable functions, Banach space Introduction In the article [16] we have defined the space L∞(a) associated with positive operator affiliated with the von Neumann algebra. Further in [20, 14] we have α α considered the commutative constructions of the limits spaces L∞(f ), L1(f ) ∗ α and L∞(f ), and found that they are total for each other in the dualities 1 α α α ∗ α hlim L1(f ), lim L∞(f )i and hlim L∞(f ), lim L∞(f )i. In this work we start to apply the same methodology for the noncommutative L∞(a) spaces. In the result we get the (LF)-spaces (the inductive limits of Frechet spaces), which are studied for example in [3, 8, 7, 10]. 1 Definitions and Notation Let B1, B2 be Banach spaces with the norms k·k1 and k·k2 such that B1 ⊂B2. We will write k·k1 ≻k·k2 if + ∃C ∈ R ∀x ∈B1 kxk2 ≤ Ckxk1. Throughout this paper we adhere to the following notation. By M we denote a von Neumann algebra that acts on a Hilbert space H with the scalar product h·, ·i. We denote its selfadjoint part by Msa, and the set of all projections in M pr pr by M . Let p ∈M and x ∈M, then by xp we denote the restriction of pxp to pH (i.e. xp := pxp|pH ), also we denote the reduction of M to pH by Mp. By h C(M) we denote the center of M. By M∗ and M∗ we denote the predual of M and its Hermitian part, respectively. If an operator x is affiliated with M then we write xηM. We denote the domain of an operator x by D(x). The adjoint operator is denoted by x∗. We denote the identity operator, the zero operator and the zero vector by 1,0 and 0 , respectively. We use standard notation for multiplication of a functional ϕ ∈ M∗ by an operator x ∈M, namely, xϕ, ϕx and xϕx denote the linear functionals y 7→ ϕ(xy), y 7→ ϕ(yx) and y 7→ ϕ(xyx), respectively. We also consider partial order for positive selfadjoint operators affiliated with M. If x is affiliated with M we denote it as xηM. For positive selfadjoint x, y η M we write x ≤ y if and only if 1 1 1 1 1 D(y 2 ) ⊂ D(x 2 ) and kx 2 fk2 ≤ky 2 fk2 for all f ∈ D(y 2 ). If for an increasing net (xj )j∈J of operators affiliated with M there exists x = sup xj , then we write xj ր x. For a positive selfadjoint operator xηM we use j∈J −1 + xλ to denote λx(λ + x) with λ ∈ R \{0}. From the Spectral theorem it + follows that the mapping λ 7→ xλ ∈M is monotone operator-valued function 1 1 1 2 and lim x f = x 2 f for all f ∈ D(x 2 ), therefore xλ ր x. For an unbounded λ→+∞ λ + x and ϕ ∈M∗ we define ϕ(x) as ϕ(x) := lim ϕ(xλ). λ→+∞ From now on a stands for a positive selfadjoint operator affiliated with M. We consider D+ a ≡{ϕ ∈M∗ | ϕ(a) < +∞}, 2 Dh D+ D+ D Dh a ≡ a − a and a ≡ linC a. D+ + Dh h D Note that if operator a is bounded, then a = M∗ , a = M∗ and a = M∗. Dh We define a seminorm k·ka on a as + kϕka := inf{ϕ1(a)+ ϕ2(a) | ϕ = ϕ1 − ϕ2; ϕ1, ϕ2 ∈ D }. Also, Theorem 2 from [7] states, that if operator a is bounded, then 1 1 2 2 kϕka = ka ϕa k. If k·ka is a norm, then we call it the a-norm. Note that the 1-norm coincides h with the restriction of the standard norm in M∗ onto M∗ . h definition 1 ([16]). By L1 (a) we denote the completion of the real normed Dh D+ space a = linR a with the norm D+ ra(ϕ) = inf{ϕ1(a)+ ϕ2(a) | ϕ = ϕ1 − ϕ2, ϕ1, ϕ2 ∈ a }, D+ + where a = {ϕ ∈M∗ | ϕ(a) < +∞}. h sa The dual of L1 (a) is (L∞(a), k·ka), where al L∞(a) ≡{x ∈ (Da) | λ ∈ R, −λa ≤ x ≤ λa} R Dh and kxka ≡ inf{λ ∈ | − λa ≤ x ≤ λa}. We identify the elements of a h with the corresponding elements in L1 (a). Further for an injective operator a sa we always assume that L∞(a) is equiped with the a-norm. 1 1 1 1 For x ∈ M we define the sesquilinear form a a 2 xa 2 on D(a 2 ) × D(a 2 ) by \1 1 1 1 the equality a 2 xa 2 (f,g) := hxa 2 f,a 2 gi. The set of all such sesquilinear forms is denoted by \1 1 Sa(M) ≡{a 2 xa 2 | x ∈ M}. sa We consider partial order on Sa(M ), such that \1 1 \1 1 a 2 xa 2 ≤ a 2 ya 2 \1 1 \1 1 1 sa if and only if a 2 xa 2 (f,f) ≤ a 2 ya 2 (f,f) for all f ∈ D(a 2 ). By Sa(M ) we 1 1 denote the seminormed space of sesquilinear forms {a 2 xa 2 | x ∈Msa} equiped \1 1 + \1 1 \1 1 \1 1 with the seminorm pa(a 2 xa 2 ) := inf{λ ∈ R | −λa 2 1a 2 ≤ a 2 xa 2 ≤ λa 2 1a 2 }. definition 2 ([1]). Let (X0,X1) be the pair of Banach spaces. For t > 0 and x ∈ X0 + X1 let K(x, t; X0,X1) = inf kx0kX0 + t kx1kX1 | x = x0 + x1, x0 ∈ X0, x1 ∈ X1 . 3 By K-method of interpolation we call the construction of the space Kθ,q(X0,X1) as the linear subspace of the sum X0 + X1 such that ∞ 1/q q dt t−θK(x, t; X ,X ) < ∞. 0 1 t Z0 h definition 3. By Lp,q(a) we denote the noncommutative Lorentz space with h h p, q ∈ (1, +∞) which is the interpolation space K(p−1)/p,q(L1 (a),L∞(a)) h definition 4. By Lp (a) we denote the noncommutative Lebesgue space which h is interpolation space Lp,p(a). 2 Preliminaries For ϕ ∈ Da the equality 1 1 1 1 a 2 ϕa 2 (x) = lim ϕ(a 2 xa 2 ) with x ∈M λ→+∞ λ λ 1 1 defines the normal functional a 2 ϕa 2 ∈M∗. If an operator a is injective then \1 1 \1 1 \1 1 inf{λ | − λa 2 1a 2 ≤ a 2 xa 2 ≤ λa 2 1a 2 } = kxk for any x ∈Msa \1 1 and the latter implies that the mapping u1 : x 7→ a 2 xa 2 is an isometrical isomorphism of M onto Sa(M). For an injective operator a the mapping \1 1 u : x ∈ M 7→ a 2 xa 2 ∈ L∞(a) is an isometrical isomporphism of M onto L∞(a). Thus, L∞(a) is isometricaly −1 isomorphic to Sa(M). Further we call the isomorphism u1 u the canonical iso- morphism of Sa() 7→ M onto L∞(a) and identify the corresponding elements. t ∗ Moreover, the adjoint mapping u is an isometrical isomorphism of L∞(a) onto M∗. For an injective operator a the continue of the mapping 1 1 2 2 v : ϕ ∈ Da(⊂ L1(a)) 7→ a ϕa ∈M∗ is an isometrical isomorphism of L1(a) onto M∗ Let B1 and B2 be Banach spaces and k·k1 ≻ k·k2, then the embedding id : x ∈B1 7→ x ∈B2 is continuous. 4 3 Mixed Limits of Banach Spaces By Bk,n we denote two-indexed family of Banach spaces. Let τ k,n be the topol- k,n ogy of the norm k·kk,n which is natural norm of the Banach space B . Let k1 < k2 and n1 <n2, ki,ni ∈ N, and consider that k1,n1 k1,n2 B ⊂ B k·kk1,n1 ≻ k·kk1,n2 ∪ ∪ and f f . k2,n1 k2,n2 B ⊂ B k·kk2,n1 ≻ k·kk2,n2 Note that k1,n1 k1,n2 τ |k2 ⊃ τ |k2,n1 ∩ ∩ , k2 ,n1 k2,n2 τ ⊃ τ |n1 k2,n1 with τ|k2 ≡ τ|n1 ≡ τ|k2 ,n1 := {X ∩B | X ∈ τ}. Consider the limits k k,n k,n B = B и Bn = B n>0 [ k>\0 k k with the topology τ and the topology τn, respectively.
Load more

Recommended publications
  • [DRAFT] a Peripatetic Course in Algebraic Topology
    [DRAFT] A Peripatetic Course in Algebraic Topology Julian Salazar [email protected]• http://slzr.me July 22, 2016 Abstract These notes are based on lectures in algebraic topology taught by Peter May and Henry Chan at the 2016 University of Chicago Math REU. They are loosely chrono- logical, having been reorganized for my benefit and significantly annotated by my personal exposition, plus solutions to in-class/HW exercises, plus content from read- ings (from May’s Finite Book), books (e.g. May’s Concise Course, Munkres’ Elements of Algebraic Topology, and Hatcher’s Algebraic Topology), Wikipedia, etc. I Foundations + Weeks 1 to 33 1 Topological notions3 1.1 Topological spaces.................................3 1.2 Separation properties...............................4 1.3 Continuity and operations on spaces......................4 2 Algebraic notions5 2.1 Rings and modules................................6 2.2 Tensor products..................................7 3 Categorical notions 11 3.1 Categories..................................... 11 3.2 Functors...................................... 13 3.3 Natural transformations............................. 15 3.4 [DRAFT] Universal properties.......................... 17 3.5 Adjoint functors.................................. 20 1 4 The fundamental group 21 4.1 Connectedness and paths............................. 21 4.2 Homotopy and homotopy equivalence..................... 22 4.3 The fundamental group.............................. 24 4.4 Applications.................................... 26
    [Show full text]
  • The Fixed-Point Property for Represented Spaces Mathieu Hoyrup
    The fixed-point property for represented spaces Mathieu Hoyrup To cite this version: Mathieu Hoyrup. The fixed-point property for represented spaces. 2021. hal-03117745v1 HAL Id: hal-03117745 https://hal.inria.fr/hal-03117745v1 Preprint submitted on 21 Jan 2021 (v1), last revised 28 Jan 2021 (v2) 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. The fixed-point property for represented spaces Mathieu Hoyrup Universit´ede Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France [email protected] January 21, 2021 Abstract We investigate which represented spaces enjoy the fixed-point property, which is the property that every continuous multi-valued function has a fixed-point. We study the basic theory of this notion and of its uniform version. We provide a complete characterization of countable-based spaces with the fixed-point property, showing that they are exactly the pointed !-continuous dcpos. We prove that the spaces whose lattice of open sets enjoys the fixed-point property are exactly the countably-based spaces. While the role played by fixed-point free functions in the diagonal argument is well-known, we show how it can be adapted to fixed-point free multi-valued functions, and apply the technique to identify the base-complexity of the Kleene-Kreisel spaces, which was an open problem.
    [Show full text]
  • Admissibly Represented Spaces and Qcb-Spaces
    Admissibly Represented Spaces and Qcb-Spaces∗ Matthias Schr¨oder Abstract A basic concept of Type Two Theory of Effectivity (TTE) is the notion of an admissibly represented space. Admissibly represented spaces are closely related to qcb-spaces. The latter form a well-behaved subclass of topological spaces. We give a survey of basic facts about Type Two Theory of Effectivity, admissibly represented spaces, qcb-spaces and effective qcb-spaces. Moreover, we discuss the relationship of qcb-spaces to other categories relevant to Computable Analysis. 1 Introduction Computable Analysis investigates computability on real numbers and related spaces. Type Two Theory of Effectivity (TTE) constitutes a popular approach to Com- putable Analysis, providing a rigorous computational framework for non-discrete spaces with cardinality of the continuum (cf. [48, 49]). The basic tool of this frame- work are representations. A representation equips the objects of a given space with names, giving rise to the concept of a represented space. Computable functions be- tween represented spaces are those which are realized by a computable function on the names. The ensuing category of represented spaces and computable functions enjoys excellent closure properties. Any represented space is equipped with a natural topology, turning it into a arXiv:2004.09450v1 [cs.LO] 20 Apr 2020 qcb-space. Qcb-spaces form a subclass of topological spaces with a remarkably rich structure. For example it is cartesian closed, hence products and function spaces can be formed. Admissibility is a notion of topological well-behavedness for representations. The category of admissibly represented spaces and continuously realizable functions is equivalent to the category QCB0 of qcb-spaces with the T0-property.
    [Show full text]
  • The Topology of Path Component Spaces
    The topology of path component spaces Jeremy Brazas October 26, 2012 Abstract The path component space of a topological space X is the quotient space of X whose points are the path components of X. This paper contains a general study of the topological properties of path component spaces including their relationship to the zeroth dimensional shape group. Path component spaces are simple-to-describe and well-known objects but only recently have recieved more attention. This is primarily due to increased qtop interest and application of the quasitopological fundamental group π1 (X; x0) of a space X with basepoint x0 and its variants; See e.g. [2, 3, 5, 7, 8, 9]. Recall qtop π1 (X; x0) is the path component space of the space Ω(X; x0) of loops based at x0 X with the compact-open topology. The author claims little originality here but2 knows of no general treatment of path component spaces. 1 Path component spaces Definition 1. The path component space of a topological space X, is the quotient space π0(X) obtained by identifying each path component of X to a point. If x X, let [x] denote the path component of x in X. Let qX : X π0(X), q(x) = [x2] denote the canonical quotient map. We also write [A] for the! image qX(A) of a set A X. Note that if f : X Y is a map, then f ([x]) [ f (x)]. Thus f determines a well-defined⊂ function f !: π (X) π (Y) given by⊆f ([x]) = [ f (x)].
    [Show full text]
  • Arxiv:2004.02357V2 [Econ.TH] 22 Jun 2021 Final Topology for Preference
    Final topology for preference spaces Pablo Schenone∗ June 23, 2021 Abstract We say a model is continuous in utilities (resp., preferences) if small per- turbations of utility functions (resp., preferences) generate small changes in the model’s outputs. While similar, these two concepts are equivalent only when the topology satisfies the following universal property: for each continuous mapping from preferences to model’s outputs there is a unique mapping from utilities to model’s outputs that is faithful to the preference map and is continuous. The topologies that satisfy such a universal prop- erty are called final topologies. In this paper we analyze the properties of the final topology for preference sets. This is of practical importance since most of the analysis on continuity is done via utility functions and not the primitive preference space. Our results allow the researcher to extrapolate continuity in utility to continuity in the underlying preferences. Keywords: decision theory, topology JEL classification: C02, D5, D01 arXiv:2004.02357v2 [econ.TH] 22 Jun 2021 ∗I wish to thank Kim Border, Laura Doval, Federico Echenique, Mallesh Pai, Omer Tamuz, and participants of the LA theory fest for insightful discussions that helped shape this paper. All remaining errors are, of course, my own. 1 1 Introduction All economic models may be thought of as a mapping from exogenous variables to endogenous variables (henceforth, model outputs or outcomes). For example, in classical demand theory, the exogenous variables are the market prices, the consumer’s wealth, and the consumer’s preferences; the model output is the con- sumer’s demand function.
    [Show full text]
  • A Topological View on Algebraic Computation Models
    © 2017, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/ Accepted Manuscript A topological view on algebraic computation models Eike Neumann, Arno Pauly PII: S0885-064X(17)30076-6 DOI: http://dx.doi.org/10.1016/j.jco.2017.08.003 Reference: YJCOM 1338 To appear in: Journal of Complexity Received date : 4 March 2016 Accepted date : 2 August 2017 Please cite this article as: E. Neumann, A. Pauly, A topological view on algebraic computation models, Journal of Complexity (2017), http://dx.doi.org/10.1016/j.jco.2017.08.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As aserviceto our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. A topological view on algebraic computation models Eike Neumanna, Arno Paulyb,c,1 aAston University, Birmingham, UK bUniversity of Cambridge, Cambridge, UK cBirmingham University, Birmingham, UK Abstract We investigate the topological aspects of some algebraic computation mod- els, in particular the BSS-model. Our results can be seen as bounds on how different BSS-computability and computability in the sense of computable anal- ysis can be. The framework for this is Weihrauch reducibility. As a conse- quence of our characterizations, we establish that the solvability complexity index is (mostly) independent of the computational model, and that there thus is common ground in the study of non-computability between the BSS and TTE setting.
    [Show full text]
  • Arxiv:Math/0603537V3 [Math.GR] 16 Jul 2008
    Final Group Topologies, Kac-Moody Groups and Pontryagin Duality Helge Gl¨ockner, Ralf Gramlich and Tobias Hartnick Abstract. We study final group topologies and their relations to compactness properties. In particular, we are interested in situations where a colimit or direct limit is locally compact, a kω-space, or locally kω. As a first application, we show that unitary forms of complex Kac-Moody groups can be described as the colimit of an amalgam of subgroups (in the category of Hausdorff topological groups, and the category of kω-groups). Our second application concerns Pontryagin duality theory for the classes of almost metrizable topological abelian groups, resp., locally kω topological abelian groups, which are dual to each other. In particular, we explore the relations between countable projective limits of almost metrizable abelian groups and countable direct limits of locally kω abelian groups. Introduction Given a group G and a family (fi)i∈I of maps fi : Xi → G from certain topological spaces to G, there exists a finest group topology on G making all of the maps fi continuous, the so-called final group topology with respect to the family (fi)i∈I . Such topologies arise naturally in connection with colimits of topological groups (notably, direct limits), which carry the final group topology with respect to the family of limit maps. Although a final group topology O always exists, it may be quite elusive in the sense that it may not be clear at all how one could check whether a given subset U ⊆ G belongs to O. For example, consider an ascending sequence G1 ⊆ G2 ⊆ ··· of topological groups such that all inclusion maps are continuous homomorphisms.
    [Show full text]
  • Time-Varying Vector Fields and Their Flows 3
    Springer Briefs in Mathematics Springer-Verlag, 2014, ISBN 978-3-319-10138-5 http://dx.doi.org/10.1007/978-3-319-10138-5 Time-varying vector fields and their flows Saber Jafarpour∗ Andrew D. Lewisy 26/01/2014 Last updated: 12/06/2014 Abstract Time-varying vector fields on manifolds are considered with measurable time de- pendence and with varying degrees of regularity in state; e.g., Lipschitz, finitely dif- ferentiable, smooth, holomorphic, and real analytic. Classes of such vector fields are described for which the regularity of the flow's dependence on initial condition matches the regularity of the vector field. It is shown that the way in which one characterises these classes of vector fields corresponds exactly to the vector fields, thought of as being vector field valued functions of time, having measurability and integrability properties associated with appropriate topologies. For this reason, a substantial part of the de- velopment is concerned with descriptions of these appropriate topologies. To this end, geometric descriptions are provided of locally convex topologies for Lipschitz, finitely differentiable, smooth, holomorphic, and real analytic sections of vector bundles. In all but the real analytic case, these topologies are classically known. The description given for the real analytic topology is new. This description allows, for the first time, a characterisation of those time-varying real analytic vector fields whose flows depend real analytically on initial condition. Keywords. Time-varying vector fields, topologies for spaces of vector fields, real ana- lyticity AMS Subject Classifications (2010). 32C05, 34A12, 46E10 Contents 1. Introduction2 1.1 Motivation. .2 1.2 An outline of the monograph.
    [Show full text]
  • General Topology Summer Term 2016
    General Topology Summer Term 2016 Michael Kunzinger [email protected] Universit¨atWien Fakult¨atf¨urMathematik Oskar-Morgenstern-Platz 1 A-1090 Wien Preface These are lecture notes for a four hour advanced course on general topology. They assume familiarity with the foundations of the subject, as taught in the two-hour introductory course offered at our faculty. In fact, a number of topics from the introductory course will be repeated here to keep prerequisites minimal. Based on this, detailed proofs are supplied for all results. Nevertheless, the approach taken is rather advanced and theory-oriented, and the overall style is in the Bourbaki spirit (which the subject matter lends itself to quite naturally). Throughout, we mainly follow the standard text [5], with occasional input from other sources (mainly [1] and [3]). Michael Kunzinger, summer term 2016 i Contents Preface i 1 New spaces from old 1 1.1 Subspaces and products . 1 1.2 Initial topologies . 4 1.3 Final topology, quotient topology . 6 1.4 Identification topology, gluing of topological spaces . 8 1.5 Manifolds and topological groups . 10 2 Filters and convergence 13 2.1 Nets . 13 2.2 Filters . 14 2.3 Convergence . 19 3 Separation properties 25 3.1 Separation axioms . 25 3.2 Inheritability of separation properties . 29 3.3 Extension by continuity . 32 4 Normal spaces 35 4.1 Urysohn's lemma . 35 4.2 Extension of continuous maps . 38 4.3 Locally finite systems and partitions of unity . 40 5 Compactness 43 5.1 Compact spaces . 43 5.2 Locally compact spaces .
    [Show full text]
  • Math 111 Final (Topology) 2006 Fall Ali Nesin As You Can See from The
    Math 111 Final (Topology) 2006 Fall Ali Nesin As you can see from the points attached to each question, the questions are not difficult at all; they are in fact rather easy. The only difficulty lies in grasping the concepts. Once you understand what the concepts are saying, the rest will be easy. Please take your time and read the definitions carefully, find your own examples. Do not try to do all the questions in order to amass points because I may take away points for absurd answers. Write carefully, clearly and legibly. Make full sentences with at least one subject and a verb. Do not use symbols such as ⇒, ∀. Let X be a set. Let τ ⊆ ℘(X) be such that T1. ∅ ∈ τ and X ∈ τ. T2. If U, V ∈ τ then U ∩ V ∈ τ. (I.e. τ is closed under finite intersections). T3. If σ ⊆ τ, then ∪ σ ∈ τ. (I.e. τ is closed under arbitrary unions). Such a τ is called a topology on X. The elements of τ are called open subsets of X (for the topology τ). The pair ( X, τ) is called a topological space . Part I. 1. Let τ be the set of cofinite subsets of X together with ∅. Show that τ is a topology on X. This is called the cofinite topology on X. (4 pts.) Answer: By definition ∅ ∈ τ. Since Xc = ∅ and ∅ is finite, X is cofinite, so X ∈ τ. Thus T1 is satisfied. Now we check T2. Let U, V ∈ τ. If U or V is ∅, then U ∩ V = ∅ ∈ τ. Assume now that neither U nor V is ∅.
    [Show full text]
  • Watts Spaces and Smooth Maps
    Watts Spaces and Smooth Maps B. Virgin Abstract - A new category is described, which generalizes a select variety of categories having smooth mappings as their class of morphisms, those like the C1 manifolds and C1 mappings between them. Categorical embeddings are produced to justify this claim of generalization. Theorems concerning the equivalence of smooth and continuous versions of different separation axioms are proved following the categorical discussion. These are followed by a generalization of Whitney's approximation theorem, a smooth version of the Tietze extension theorem, and a sufficient condition to guarantee that the connected com- ponents of these spaces are smoothly path-connected. Keywords : Souriau (or diffeological) space; Sikorski (or differential) space; Fr¨olicher space; smoothness; separation; approximation; extension Mathematics Subject Classification (2020) : 57R55 ; 58A40 1 Introduction The subject of differential topology began with the study of finite dimensional smooth manifolds, objects which retained many of the features of the euclidean spaces studied classically. These incredibly specialized spaces have more than enough structure to allow for the smoothness of maps between them to be defined, and for many important theorems in elementary calculus to be extended in a natural way. They also possess enough structure to ensure many other desirable properties, like the equivalence of all notion of tangent space, connectedness implying smooth path connectedness, and continuous homotopy implying smooth homotopy for smooth maps. These spaces are also incredibly relevant to the physical sciences, being central not only to General Relativity, but also to the modern gauge-symmetry approach to Quantum Field Theory. Ever since the inception of smooth manifolds, it was understood that there were larger categories at work, as manifolds with boundaries were quickly introduced alongside the original spaces, followed shortly thereafter by those with a boundary and corners.
    [Show full text]
  • On Completions of LB-Spaces of Moscatelli Type (Dedicated to the Memory of Walter Roelcke, Doctoral Promotor of the first Author)
    RAC Rev. R. Acad.SAM Cien. Serie A. Mat. VOL. 102 (2), 2008, pp. 205–209 Analisis´ Matematico´ / Mathematical Analysis On Completions of LB-spaces of Moscatelli Type (Dedicated to the Memory of Walter Roelcke, doctoral promotor of the first author) Susanne Dierolf and Phillip Kuß Abstract. We first present a class of LF-spaces, extending the class of LF-spaces of Moscatelli type, for which regularity implies completeness. Then we utilize the obtained results to describe the completions of LB-spaces of Moscatelli type. In particular, we prove that the completions of LB-spaces of that type are again LB-spaces. Sobre las completaciones de espacios LB de tipo Moscatelli Resumen. Presentamos primero una clase de espacios LF, que extiende los espacios LF de tipo Moscatelli, para la cual la regularidad del l´ımite inductivo implica la completitud. A continuacion´ uti- lizamos los resultados obtenidos para describir la completacion´ de los espacios LB de tipo Moscatelli usual. En particular, demostramos que la completacion´ de un espacio LB de ese tipo es tambien´ un espacio LB. LB- and LF-spaces of Moscatelli type have been thoroughly investigated since the late eighties, and have acted as a rich source of examples and counterexamples in various contexts (see e.g. [1,2,3]). In particular it is known for almost twenty years that LF-spaces of that class are complete, if they are regular. On the other hand, no information about the complete hull of LB-space of that type had been obtained. In view of the following well-known implication: “If it is true that every regular LB-space is complete, then the completion of every LB-space is again an LB-space” a description of the complete hulls of certain LB- spaces is desirable.
    [Show full text]