DOCSLIB.ORG

On Continuity of Functions Between Vector Metric Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Continuity of Functions Between Vector Metric Spaces Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 753969, 6 pages http://dx.doi.org/10.1155/2014/753969 Research Article On Continuity of Functions between Vector Metric Spaces Cüneyt Çevik Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, 06500 Ankara, Turkey Correspondence should be addressed to Cuneyt¨ C¸ evik; [email protected] Received 17 August 2014; Accepted 8 October 2014; Published 12 November 2014 Academic Editor: Han Ju Lee Copyright © 2014 Cuneyt¨ C¸ evik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce vectorial and topological continuities for functions defined on vector metric spaces and illustrate spaces of such functions. Also, we describe some fundamental classes of vector valued functions and extension theorems. 1. Introduction and Preliminaries vector space has started by Zabrejko in [8]. The distance map in the sense of Zabrejko takes values from an ordered vector Let be a Riesz space. If every nonempty bounded above space. We use the structure of lattice with the vector metrics (countable) subset of has a supremum, then is called having values in Riesz spaces; then we have new results as Dedekind (-) complete.Asubsetin is called order bounded mentioned above. This paper is on some concepts and related if it is bounded both above and below. We write ↓if () results about continuity in vector metric spaces. is a decreasing sequence in such that inf =.TheRiesz −1 space is said to be Archimedean if ↓0holds for every Definition 1. Let be a nonempty set and let be a Riesz :×→ ∈+.Asequence() is said to be -convergent (or order space. The function is said to be a vector convergent)to if there is a sequence () in such that ↓0 metric (or -metric) if it satisfies the following properties: and |−| ≤ for all ,where|| = ∨(−) for any ∈.We (vm1) (, ) =0 if and only if =, will denote this order convergence by → .Furthermore,a sequence () is said to be -Cauchy if there exists a sequence (vm2) (, ) ≤ (, ) +(,) for all , ,. ∈ () in such that ↓0and | −+|≤ for all and . The Riesz space is said to be -Cauchy complete if every Also the triple (, , ) is said to be vector metric space. -Cauchy sequence is -convergent. (, , ) An operator :between → two Riesz spaces is Definition 2. Let be a vector metric space. positive if () ≥0 for all ≥0.Theoperator is said ( ) to be order bounded if it maps order bounded subsets of (a) A sequence in is vectorially convergent (or is ∈ to order bounded subsets of .Theoperator is called - -convergent) to some ,ifthereisasequence () in such that ↓0and (,)≤ for all . order continuous if →0 in implies () →0 in , .Every-order continuous operator is order bounded. If We will denote this vectorial convergence by → (∨)= ()∨()for all , ,theoperator ∈ is . a lattice homomorphism. For further information about Riesz (b) A sequence () in is called -Cauchy sequence spaces and the operators on Riesz spaces, we refer to [1, 2]. whenever there exists a sequence () in such that In [3], a vector metric space is defined with a distance map ↓0and (,+)≤ for all and . having values in a Riesz space, and some results in metric space theory are generalized to vector metric space theory. (c) The vector metric space is called -complete if each Some fixed point theorems in vector metric spaces are given -Cauchy sequence in is -convergenttoalimitin in [3–7].Actually,thestudyofmetricspaceshavingvalueona . 2 Journal of Function Spaces One of the main goals of this paper is to demonstrate the Corollary 5. For a function :between → two vector properties of functions on vector metric spaces in a context metric spaces (, , ) and (,,) the following statements more general than the continuity in metric analysis. Hence, hold. the properties of Riesz spaces will be the tools for the study of (a) If is Dedekind -complete and is vectorially contin- continuity of vector valued functions. The result we get from (( ), ()) ↓0 ( ,)↓0 here is continuity in general sense, an order property rather uous, then whenever . than a topological feature. (b) If is Dedekind -complete and ((), ()) ↓ In Section 2, we consider two types of continuity on vec- 0 whenever (,) ↓, 0 then the function is tor metric spaces. This approach distinguishes continuities vectorially continuous. vectorially and topologically. Moreover, vectorial continuity (c) Suppose that and are Dedekind -complete. Then, examples are given and the relationship between vectorial the function is vectorially continuous if and only if continuity of a function and its graph is demonstrated. In ((), ()) ↓0 whenever (,)↓0. Section 3, equivalent vector metrics, vectorial isometry, vec- , torial homeomorphism definitions, and examples related to ( ,) ↓ 0 → these concepts are given. In Section 4,uniformcontinuityis Proof. (a) If ,then .Bythevectorial continuity of the function ,thereisasequence() in discussed and some extension theorems for functions defined ↓0 (( ), ()) ≤ on vector metric spaces are given. Finally, in Section 5, such that and for all .Since is Dedekind -complete, ((), ()) ↓0 holds. a uniform limit theorem on a vector metric space is given, , and the structure of vectorial continuous function spaces is (b) Let →in . Then there is a sequence () in demonstrated. such that ↓0and (,) ≤ for all .Since is Dedekind -complete, (,)↓ 0holds. By the hypothesis, , 2. Topological and Vectorial Continuity ((), ()),andso ↓0 () → () in . (c) Proof is a consequence of (a) and (b). We now introduce two types of continuity in vector metric , spaces. For two elements and in a Riesz space, we write (,,) → < ≤ ≠ Example 6. Let be a vector metric space. If to indicate that but . , and →,then(,) → (,;thatis,thevector ) (, , ) (,, ) 2 Definition 3. Let and be vector metric metric map from to is vectorially continuous. Here, ∈ 2 ̃ spaces, and let . is equipped with the -valued vector metric defined (,̃ ) = ( , )+(, ) =(, ), = (a) A function :is → said to be topologically as 1 2 1 2 for all 1 1 ( , )∈2 continuous at if for every >0in there exists 2 2 ,and is equipped with the absolute valued |⋅| some in such that ((), ()) < whenever vector metric . ∈ (, ) < and .Thefunction is said (, , ) to be topologically continuous if it is topologically We recall that a subset of a vector metric space is called -closed (or vector closed) whenever ()⊆and continuous at each point of . , →imply ∈. (b) A function :is → said to be vectorially , , continuous at if →in implies () → Theorem 7. Let (, , ) and (,,)be vector metric spaces. () in .Thefunction is said to be vectorially If a function :is → vectorially continuous, then for −1 continuous if it is vectorially continuous at each point every -closed subset of the set () is -closed in . of . −1 Proof. For any ∈ (), there exists a sequence () in Theorem 4. (, , ) (,,) , Let and be vector metric spaces −1() → : → such that . Since the function is vectorially where is Archimedean. If a function is , topologically continuous, then is vectorially continuous. continuous, () → ().Buttheset is -closed, so ∈ −1 −1 ().Then,theset () is -closed. , Proof. Suppose that →. Then there exists a sequence If and are two Riesz spaces, then ×is also a Riesz () in such that ↓0and (,)≤ for all .Let be space with coordinatewise ordering defined as any nonzero positive element in . By topological continuity =((1/); ) of at , there exist elements in such that (1,1)≤(2,2)⇐⇒1 ≤2,1 ≤2 (1) (, )< implies ((), ( )) < (1/) for all .Then = ∨ ( ,) ≤ there exist elements in such that for all (1,1), (2,2)∈×. The Riesz space ×is < implies ((), ()) < (1/) for all .Since is a vector metric space, equipped with the biabsolute valued , vector metric |⋅|defined as Archimedean, (1/).Hence, ↓0 () → (). |−| = (1 −2 , 1 −2) (2) Vectorially continuous functions have a number of nice characterizations. for all =(1,1), =(2,2)∈×. Journal of Function Spaces 3 Let and be two vector metrics on which are - function ℎ from ×to defined by ℎ(, ) = |() −()| valued and -valued, respectively. Then the map defined for all ∈, ∈is vectorially continuous with the ×- as valued product vector metric and the absolute valued vector metric |⋅|. (,) = ( (,) ,(,)) (3) (c) If : (, , ) → (, ,) and : (,,) → , ∈ × (,, ) are vectorially continuous functions, then the func- for all is an -valued vector metric on .Wewill ℎ × × ℎ(, ) = ((), ()) call double vector metric. tion from to defined by for all ∈, ∈is vectorially continuous with the ×- × Example 8. Let (,,) and (, , ) be vector metric valued and -valued product vector metrics. spaces. We have the next proposition for any product vector (i) Suppose that :and → :are → metric. vectorially continuous functions. Then the function ℎ from to ×defined by ℎ() = ((), ()) for all Proposition 12. Let ()=(,) be a sequence in ( × ∈ ,× is vectorially continuous with the double vector , , ×) and let = (,)∈.Then, × → metric and the biabsolute valued vector metric |⋅|. , , ifandonlyif →and →. (ii) Let :×→and :×→be the ℎ: → × projection maps. Any function can be Now let us give relevance between vectorial continuity of ℎ() = ((), ()) ∈ = written as for all where a function and closeness of its graph. ℎ and =ℎ.Ifℎ is vectorially continuous, so are and since and are vectorially continuous. Corollary 13. Let (,,) and
Load more

Recommended publications
  • Math 395: Category Theory Northwestern University, Lecture Notes
    Math 395: Category Theory Northwestern University, Lecture Notes Written by Santiago Can˜ez These are lecture notes for an undergraduate seminar covering Category Theory, taught by the author at Northwestern University. The book we roughly follow is “Category Theory in Context” by Emily Riehl. These notes outline the specific approach we’re taking in terms the order in which topics are presented and what from the book we actually emphasize. We also include things we look at in class which aren’t in the book, but otherwise various standard definitions and examples are left to the book. Watch out for typos! Comments and suggestions are welcome. Contents Introduction to Categories 1 Special Morphisms, Products 3 Coproducts, Opposite Categories 7 Functors, Fullness and Faithfulness 9 Coproduct Examples, Concreteness 12 Natural Isomorphisms, Representability 14 More Representable Examples 17 Equivalences between Categories 19 Yoneda Lemma, Functors as Objects 21 Equalizers and Coequalizers 25 Some Functor Properties, An Equivalence Example 28 Segal’s Category, Coequalizer Examples 29 Limits and Colimits 29 More on Limits/Colimits 29 More Limit/Colimit Examples 30 Continuous Functors, Adjoints 30 Limits as Equalizers, Sheaves 30 Fun with Squares, Pullback Examples 30 More Adjoint Examples 30 Stone-Cech 30 Group and Monoid Objects 30 Monads 30 Algebras 30 Ultrafilters 30 Introduction to Categories Category theory provides a framework through which we can relate a construction/fact in one area of mathematics to a construction/fact in another. The goal is an ultimate form of abstraction, where we can truly single out what about a given problem is specific to that problem, and what is a reflection of a more general phenomenom which appears elsewhere.
    [Show full text]
  • Chapter 5 Euclidean Spaces
    Chapter 5 Euclidean Spaces 5.1 Inner Products, Euclidean Spaces The framework of vector spaces allows us deal with ratios of vectors and linear combinations, but there is no way to express the notion of length of a line segment or to talk about orthogonality of vectors. AEuclideanstructurewillallowustodealwithmetric notions such as orthogonality and length (or distance). First, we define a Euclidean structure on a vector space. 249 250 CHAPTER 5. EUCLIDEAN SPACES Definition 5.1. ArealvectorspaceE is a Euclidean space iffit is equipped with a symmetric bilinear form ϕ: E E R which is also positive definite,which means× that → ϕ(u, u) > 0, for every u =0. More explicitly, ϕ: E E R satisfies the following axioms: × → ϕ(u1 + u2,v)=ϕ(u1,v)+ϕ(u2,v), ϕ(u, v1 + v2)=ϕ(u, v1)+ϕ(u, v2), ϕ(λu, v)=λϕ(u, v), ϕ(u, λv)=λϕ(u, v), ϕ(u, v)=ϕ(v, u), u =0 impliesthat ϕ(u, u) > 0. The real number ϕ(u, v)isalsocalledtheinner product (or scalar product) of u and v. 5.1. INNER PRODUCTS, EUCLIDEAN SPACES 251 We also define the quadratic form associated with ϕ as the function Φ: E R+ such that → Φ(u)=ϕ(u, u), for all u E. ∈ Since ϕ is bilinear, we have ϕ(0, 0) = 0, and since it is positive definite, we have the stronger fact that ϕ(u, u)=0 iff u =0, that is Φ(u)=0iffu =0. Given an inner product ϕ: E E R on a vector space E,wealsodenoteϕ(u, v)by× → u v, or u, v , or (u v), · | and Φ(u)by u .
    [Show full text]
  • Some Geometric and Dynamical Properties of the Urysohn Space
    Some geometric and dynamical properties of the Urysohn space Julien Melleray Abstract This is a survey article about the geometry and dynamical properties of the Urysohn space. Most of the results presented here are part of the author's Ph.D thesis and were published in the articles [Me1], [Me2] and [Me3]; a few results are new, most notably the fact that Iso(U) is not divisible. Introduction. This paper has two main objectives: presenting the author's results about the Urysohn space and its isometry group in a unied setting, and providing an introduction to the techniques and methods that are commonly used to study this space. Hopefully the paper is suciently self-contained to be of use to people who haven't worked on the Urysohn space before, and still covers basic material fast enough not to bore people who already know it. Urysohn's universal metric space U was characterized in Urysohn's original paper [Ur] as being, up to isometry, the unique Polish metric space with the following two properties: - Given any two isometric nite metric subsets A, A0 ⊂ U, and any isometry ϕ: A → A0, there exists an isometry ϕ˜ of U which extends ϕ; - Any separable metric space is isometric to a subspace of U. The rst property is now called ω-homogeneity (or ultrahomogeneity) the sec- ond is called universality. There are other examples of ω-homeogenous Polish metric spaces, for example the Hilbert space; similarly, there are other uni- versal Polish metric spaces, the best-known example being perhaps C([0, 1]).
    [Show full text]
  • Arxiv:1102.4381V1 [Math.MG] 22 Feb 2011 N DMS-0653439
    RIGIDITY OF SCHOTTKY SETS MARIO BONK, BRUCE KLEINER, AND SERGEI MERENKOV DEDICATED TO THE MEMORY OF JUHA HEINONEN Abstract. We call a complement of a union of at least three disjoint (round) open balls in the unit sphere Sn a Schottky set. We prove that every quasisymmetric homeomorphism of a Schottky set of spherical measure zero to another Schottky set is the restriction of a M¨obius transformation on Sn. In the other direction we show that every Schottky set in S2 of positive measure admits non-trivial quasisymmetric maps to other Schottky sets. These results are applied to establish rigidity statements for con- vex subsets of hyperbolic space that have totally geodesic bound- aries. 1. Introduction Let Sn denote the n-dimensional unit sphere in Rn+1 equipped with the restriction of the Euclidean metric. A Schottky set is a subset of Sn whose complement is a union of at least three disjoint metric (i.e., round) open balls. We impose the requirement that a Schottky set has at least three distinct open balls as complementary components to rule out cases that are easy to analyze for the type of problems we consider. Each Schottky set is endowed with the induced metric from Sn. Let f : X → Y be a homeomorphism between two metric spaces (X,dX ) and (Y,dY ). The map f is called η-quasisymmetric, where arXiv:1102.4381v1 [math.MG] 22 Feb 2011 η : [0, ∞) → [0, ∞) is a homeomorphism, if d (f(x), f(y)) d (x, y) Y ≤ η X dY (f(x), f(z)) dX(x, z) for every triple of distinct points x, y, z ∈ X.
    [Show full text]
  • APPROACHING METRIC DOMAINS 3 Which Is Equivalent to O(Y) ⊆ O(X), Hence It Is the Dual of the Specialisation Order
    APPROACHING METRIC DOMAINS GONC¸ALO GUTIERRES AND DIRK HOFMANN Abstract. In analogy to the situation for continuous lattices which were introduced by Dana Scott as precisely the injective T0 spaces via the (nowadays called) Scott topology, we study those metric spaces which correspond to injective T0 approach spaces and characterise them as precisely the continuous lattices equipped with an unitary and associative [0, ∞]-action. This result is achieved by a thorough analysis of the notion of cocompleteness for approach spaces. Introduction Domain theory is generally concerned with the study of ordered sets admitting certain (typically up- directed) suprema and a notion of approximation, here the latter amounts to saying that each element is a (up-directed) supremum of suitably defined “finite” elements. From a different perspective, domains can be viewed as very particular topological spaces; in fact, in his pioneering paper [Scott, 1972] Scott introduced the notion of continuous lattice precisely as injective topological T0 space. Yet another point of view was added in [Day, 1975; Wyler, 1985] where continuous lattices are shown to be precisely the algebras for the filter monad. Furthermore, suitable submonads of the filter monad have other types of domains as algebras (for instance, continuous Scott domains [Gierz et al., 2003] or Nachbin’s ordered compact Hausdorff spaces [Nachbin, 1950]), and, as for continuous lattices, these domains can be equally seen as objects of topology and of order theory. This interplay between topology and algebra is very nicely explained in [Escard´oand Flagg, 1999] where, employing a particular property of monads of filters, the authors obtain “new proofs and [.
    [Show full text]
  • Coordinate Systems and Bounded Isomorphisms
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications, Department of Mathematics Mathematics, Department of 12-20-2005 COORDINATE SYSTEMS AND BOUNDED ISOMORPHISMS David R. Pitts David R. Pitts Follow this and additional works at: https://digitalcommons.unl.edu/mathfacpub Part of the Applied Mathematics Commons, and the Mathematics Commons This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Faculty Publications, Department of Mathematics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. J. OPERATOR THEORY © Copyright by THETA, 2008 59:2(2008), 359–416 COORDINATE SYSTEMS AND BOUNDED ISOMORPHISMS ALLAN P. DONSIG and DAVID R. PITTS Communicated by Kenneth R. Davidson ∗ ABSTRACT. For a Banach D-bimodule M over an abelian unital C -algebra D, we define E1(M) as the collection of norm-one eigenvectors for the dual action of D on the Banach space dual M#. Equip E1(M) with the weak*-topology. We develop general properties of E1(M). It is properly viewed as a coordinate system for M when M ⊆ C, where C is a unital C∗-algebra containing D as a regular MASA with the extension property; moreover, E1(C) coincides with Kumjian’s twist in the context of C∗-diagonals. We identify the C∗-envelope of a subalgebra A of a C∗-diagonal when D ⊆ A ⊆ C. For triangular subal- gebras, each containing the MASA, a bounded isomorphism induces an al- gebraic isomorphism of the coordinate systems which can be shown to be continuous in certain cases.
    [Show full text]
  • Extensions of Bundles of C*-Algebras Are Functo- Rial
    Extensions of Bundles of C*-algebras Jeremy Steeger Benjamin H. Feintzeig [email protected] [email protected] Department of Philosophy Department of Philosophy University of Washington University of Washington May 26, 2021 Abstract Bundles of C*-algebras can be used to represent limits of physical theories whose algebraic structure depends on the value of a parameter. The primary example is the ~ → 0 limit of the C*-algebras of physical quantities in quantum theories, represented in the framework of strict deformation quantization. In this paper, we understand such limiting procedures in terms of the extension of a bundle of C*-algebras to some limiting value of a parameter. We prove existence and uniqueness results for such extensions. Moreover, we show that such extensions are functorial for the C*-product, dynamical automorphisms, and the Lie bracket (in the ~ → 0 case) on the fiber C*-algebras. Contents 1 Introduction 2 2 Motivation from Quantization 3 3 UC b Bundles 5 4 Existence and Uniqueness of Extensions 7 arXiv:2102.11993v2 [math.OA] 24 May 2021 5 Functoriality 11 5.1 C*-algebraicstructure . ........ 12 5.2 Dynamicalstructure .............................. ...... 14 5.3 Liebracket ...................................... 16 6 Conclusion 19 A From Quantization to Bundles 20 B Continuity and Uniform Continuity 22 1 1 Introduction In this paper, we are interested in limiting procedures in which the algebraic structure of a theory varies depending on the numerical value of some parameter. In many cases, the physical quantities of such a theory can be represented by a C*-algebra of quantities, and the corresponding limit can be represented by a bundle of C*-algebras [12, 18].
    [Show full text]
  • Approaching Metric Domains
    APPROACHING METRIC DOMAINS GONC¸ALO GUTIERRES AND DIRK HOFMANN Abstract. In analogy to the situation for continuous lattices which were introduced by Dana Scott as precisely the injective T0 spaces via the (nowadays called) Scott topology, we study those metric spaces which correspond to injective T0 approach spaces and characterise them as precisely the continuous lattices equipped with a unitary and associative [0; 1]-action. This result is achieved by a detailed analysis of the notion of cocompleteness for approach spaces. Introduction Domain theory is generally concerned with the study of ordered sets admitting certain (typically up- directed) suprema and a notion of approximation, here the latter amounts to saying that each element is a (up-directed) supremum of suitably defined “finite” elements. From a different perspective, domains can be viewed as very particular topological spaces; in fact, in his pioneering paper [Scott, 1972] Scott introduced the notion of continuous lattice precisely as injective topological T0 space. Yet another point of view was added in [Day, 1975; Wyler, 1985] where continuous lattices are shown to be precisely the algebras for the filter monad. Furthermore, suitable submonads of the filter monad have other types of domains as algebras (for instance, continuous Scott domains [Gierz et al., 2003] or Nachbin's ordered compact Hausdorff spaces [Nachbin, 1950]), and, as for continuous lattices, these domains can be equally seen as objects of topology and of order theory. This interplay between topology and algebra is very nicely explained in [Escard´oand Flagg, 1999] where, employing a particular property of monads of filters, the authors obtain \new proofs and [.
    [Show full text]
  • On the Topologies of the Exponential
    ONTHETOPOLOGIESOFTHE EXPONENTIAL annacepek damienlejay Factorization algebras haveAbstract been defined using three different topo- logies on the Ran space. We study these three different topologies on the exponential, which is the union of the Ran space and the empty configur- ation, and show that an exponential property is satisfied in each case. As a consequence we describe the weak homotopy type of the exponential Exp(X) for each topology, in the case where X is not connected. We also study these exponentials as stratified spaces and show that the metric exponential is conically stratified for a general class of spaces. As a corollary, we obtain that locally constant factorization algebras defined by Beilinson-Drinfeld are equivalent to locally constant factoriz- ation algebras defined by Lurie. 1 Exponentials3 contents1.1 Exponential functors3 1.2 The exponential functor6 2 Topologies on the exponential9 2.1 The topological exponential9 2.1.1 First steps towards exponentiability9 2.1.2 The topological exponential is not an exponential 10 2.1.3 The topological exponential is almost an exponential 11 2.1.4 The topological exponential is a restricted exponential 11 2.2 The metric exponential 12 2.3 The minimal exponential 14 2.4 Weak homotopy type of the exponentials 15 3 Interlude: minimal enclosing balls 17 arXiv:2107.11243v1 [math.AT] 23 Jul 2021 4 Stratification of the exponentials 19 4.1 The exponentials as stratified spaces 19 4.2 Cones and joins 20 4.3 Conical stratification 21 This work was supported by IBS-R003-D1. 1 Roughly speaking, a factorization algebra on a space X with values in a A symmetric monoidal category is a gadget associating to each finite subset C⊗ of points S X an object , such that ⊂ AS 2 C i I Si i I Si for every finite family I of disjoint finite At 2 ⊗ 2 A subsets S X; i ⊂ factorizationthe assignment S be continuous.
    [Show full text]
  • Rough Isometries of Order Lattices and Groups
    Rough Isometries of Order Lattices and Groups Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades “Doctor rerum naturalium” der Georg-August-Universit¨at G¨ottingen vorgelegt von Andreas Lochmann aus Bad Pyrmont G¨ottingen 2009 Referenten der Dissertation: Prof. Dr. Thomas Schick Prof. Laurent Bartholdi, Ph.D. Mitglieder der Pr¨ufungskommission: Prof. Dr. Thomas Schick Prof. Laurent Bartholdi, Ph.D. Prof. Dr. Rainer Kress Prof. Dr. Ralf Meyer Prof. Dr. Ulrich Stuhler Prof. Dr. Andreas Thom Tag der m¨undlichen Pr¨ufung: 6. August 2009 Gewidmet meinen Eltern Preface Assume you measure the distances between four points and find the following: One of the distances is 143, the distance between the other pair of points is 141, and all other distances are 99. You know that your error in measurement is not greater than 1. Is this result then compatible with the hypothesis, that the four points constitute a square in Euclidean space? Another example: You distribute five posts equidistantly along straight rails, then watch a train driving on the rails and take the times at which it passes the posts. The times you measure are: 0, 100, 202, 301, 401. However, the time the clocks show might deviate by up to 1 from the clock at the starting post. Are the measured times consistent with the assumption that the train has constant speed in Newtonian mechanics? These two puzzles are metaphors for a general problem in the application of mathematics. We have to use measurements that might contain errors to verify predictions of an ideal theory. If, for example, you find a function f between vector spaces to fulfill linearity only approximately, i.e.
    [Show full text]
  • Volumes and Areas of Lipschitz Metrics
    VOLUMES AND AREAS OF LIPSCHITZ METRICS S. V. IVANOV Abstract. We develop and generalize methods of estimating (Riemannian and Finsler) filling volumes using nonexpanding maps to Banach spaces of L1 type. For every Finsler volume functional (such as the Busemann volume or the Holmes{Thompson volume) we construct a natural extension from the class of Finsler metrics to all Lipschitz metrics and define the notion of area for Lipschitz surfaces in a Banach space. We establish a correspondence between minimal fillings and minimal surfaces in L1 type spaces. We introduce a Finsler volume functional for which Riemannian and Finsler filling volumes are equal and prove that this functional is semi-elliptic. Introduction 0.1. Motivations. This paper is motivated by filling minimality and boundary rigidity problems for Riemannian manifolds. Let (M; g) be a compact Riemannian manifold with boundary, S = @M. Denote by dg the associated distance function, dg : M × M ! R. Let d : S × S ! R be an arbitrary metric on S. We say that a compact Riemannian manifold (M; g) is a filling of a metric space (S; d) if dg(x; y) ≥ d(x; y) for all x; y 2 S. The filling volume FillVol(S; d) of (S; d) is defined by FillVol(S; d) = inffvol(M; g):(M; g) is a filling of (S; d)g; cf. [17]. (This definition makes sense only for null-cobordant manifolds S, in gen- eral one should let M range over all pseudo-manifolds or complete non-compact manifolds.) We say that (M; g) is a minimal filling if it realizes the above infimum, that is, vol(M; g) = FillVol(@M; dgjS×S).
    [Show full text]
  • Continuity in Vector Metric Spaces
    CONTINUITY IN VECTOR METRIC SPACES CUNEYT¨ C¸EVIK˙ Abstract. We introduce vectorial and topological continuities for functions defined on vector metric spaces and illustrate spaces of such functions. Also, we describe some fundamental classes of vector valued functions and extension theorems. 1. Introduction and Preliminaries Let E be a Riesz space. If every nonempty bounded above (countable) subset of E has a supremum, then E is called Dedekind (σ-)complete. A subset in E is called order bounded if it is bounded both above and below. We write an ↓ a if (an) is a decreasing sequence in E such that inf an = a. The Riesz space E is −1 said to be Archimedean if n a ↓ 0 holds for every a ∈ E+. A sequence (bn) is said to be o-convergent (or order convergent) to b if there is a sequence (an) in E such that an ↓ 0 and |bn − b| ≤ an for all n, where |a| = a ∨ (−a) for any a ∈ E. o We will denote this order convergence by bn −→ b. Furthermore, a sequence (bn) is said to be o-Cauchy if there exists a sequence (an) in E such that an ↓ 0 and |bn − bn+p|≤ an for all n and p. The Riesz space E is said to be o-Cauchy complete if every o-Cauchy sequence is o-convergent. An operator T : E → F between two Riesz spaces is positive if T (x) ≥ 0 for all x ≥ 0. The operator T is said to be order bounded if it maps order bounded subsets of E to order bounded subsets of F .
    [Show full text]