DOCSLIB.ORG

Lecture Course on Coarse Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Lecture course on coarse geometry Ulrich Bunke∗ February 8, 2021 Contents 1 Bornological spaces2 2 Coarse spaces9 3 Bornological coarse spaces 26 4 Coarse homology theories 34 5 Coarse ordinary homology 46 6 Coarse homotopy 54 7 Continuous controlled cones and homology of compact spaces 57 8 Controlled objects in additive categories 67 9 Coarse algebraic K-homology 75 10 The equivariant case 77 11 Uniform bornological coarse spaces 87 12 Topological K-theory 109 References 123 ∗Fakult¨atf¨urMathematik, Universit¨atRegensburg, 93040 Regensburg, [email protected] regensburg.de 1 1 Bornological spaces In this section we introduce the notion of a bornological space. We show that the category of bornological spaces and proper maps is cocomplete and almost cocomplete. We explain how bornologies can be constructed and how they are applied. Let X be a set. By PX we denote the power set of X. Let B be a subset of PX . Definition 1.1. B is called a bornology if it has the following properties: 1. B is closed under taking subsets. 2. B is closed under forming finite unions. S 3. B2B B = X. The elements of the bornology are called the bounded subsets of X. 2 in Definition 1.1 Definition 1.2. B is a generalized bornology if it satisfies the Conditions 1 and 2 in Definition 1.1. Thus we we get the notion of a generalized bornology by dropping Condition 3 in Definition 1.1. Remark 1.3. Let x be in X. Then the singleton fxg belongs to any bornology. Indeed, by Condition 3 there exists an element B in B such that x 2 B. Then fxg ⊆ B and hence fxg 2 B by Condition 1. If B is a generalized bornology, then a point x in X is called bounded if fxg 2 B. Otherwise it is called unbounded. A generalized bornology is a bornology if and only all points of X are bounded. If X is a generalized bornological space, then we have a disjoint decomposition X = Xb tXu into the subsets of bounded and unbounded points. Then B becomes a bornology on Xb. Remark 1.4. Let A be an abelian group and consider the abelian group AX of functions from X to A. For f in AX we let supp(f) := fx 2 X j f(x) 6= 0g be the support of f. We have supp(f + f 0) ⊆ sup(f) [ supp(f 0). If B is a generalized bornology on X, then we can consider the subset X CB(X; A) := ff 2 A j supp(f) 2 Bg : 2 X 0 We shall see that this is a subgroup of A . Indeed, if f; f are in CB(X; A), then supp(f) and supp(f) belong to B. Since supp(f + f 0) ⊆ supp(f) [ supp(f 0) we see that also 0 0 supp(f + f ) belongs to B so that f + f is in CB(X; A). Similarly, supp(−f) = supp(f) so that with f also −f belongs to CB(X; A). In this way B determines a subgroup of AX of functions with bounded support. X 0 If A is a ring, then A is also a ring and CB(X; A) is a subring since supp(ff ) ⊆ supp(f) \ supp(f 0). We now turn to examples and constructions of bornologies. Let X be a set and (Bi)i2I be a family of (generalized) bornologies on X. T Lemma 1.5. The intersection i2I Bi is a (generalized) bornology. T 0 Proof. Let B := i2I Bi. Consider B in B and let B be a subset of B. Then B 2 Bi for 0 0 0 all i in I and hence B 2 Bi for all i in I. Hence B 2 B . 0 0 0 Assume that B; B belong to B. Then B 2 Bi and B 2 Bi for all i in I. Hence B [B 2 Bi for all i in I and hence B [ B0 2 B. This finishes the case of generalized bornologies. For the case of bornologies we consider x S in X. Then fxg 2 Bi for every i in I and hence fxg 2 B. Hence x 2 B2B B. Let A be a subset of PX . Then there is a smallest bornology containing A given by \ BhAi := B ; B; A⊆B where the intersection runs over all bornologies B on X. Similarly there is a smallest generalized bornology containing A \ BhAi~ = B~ ; B; A⊆B~ where the intersection runs over all generalized bornologies B on X containing A. S We can describe BhAi explicitly. We will assume that A2A A = X. Since singletons belong to every bornology (see Example 1.3) we can enlarge A by singletons without changing BhAi in order to ensure this condition. S Lemma 1.6. Assume that A = X. Then a subset B in PX belongs to BhAi if and A2A S only if of there exists a finite family (Ai)i2I in A such that B ⊆ i2I Ai. 3 Proof. We consider the subset B0 of P of subsets B of X such that there exists a finite S family (Ai)i2I in A with B ⊆ i2I Ai. One checks that B0 is a bornology which contains A. Therefore BhAi ⊆ B0. On the other hand, B0 is contained in any other bornology which contains A. This implies that B0 ⊆ BhAi. ~ Remark 1.7. The generalized bornology BhAi has a similar description. We let Xb := S 0 A2A A. Then A generates a bornology B on Xb whose elements are described as in Lemma 1.6. We have BhAi~ = B0. Example 1.8. If X is a set, then it has the minimal bornology Bmin of finite subsets and the maximal bornology Bmax of all subsets. It has the empty generalized bornology. Example 1.9. Let X be a topological space. The set Bqc of subsets of quasi-compact subsets of X is a bornology. The set Brc of relatively quasi compact subsets (subsets whose closures are quasi compact) is a generalized bornology. It might happen that the closure of a point is not quasi compact. This can happen of X is not Hausdorff. If we replace the condition \quasi compact" by \compact", then in general we do not get a bornology since the union of two compact subsets is not necessarily compact. Again this can happen since compactness includes the condition of being Hausdorff and a union of two Hausdorff subsets need not be Hausdorff. Example 1.10. Let d be a quasi-metric (infinite distances are allowed) on X. Then the metrically bounded subsets generate a bornology Bd. Note that we must say \generate" since the union of two bounded sets might be unbounded. But if d is a metric, then the bounded sets form a bornology since Condition 2 follows from the triangle inequality for the metric. The bornology Bd is generated by the set of balls fB(x; r) j x 2 X r 2 [0; 1)g. Example 1.11. Let Y be a topological space, A be a subset, and X := Y n A. Let B := fZ ⊆ X j Z¯ \ A = ;g. This is a generalized bornology. If Y is Hausdorff then it is a bornology. Example 1.12. Recall that a filter on a set X is a subset F of PX with the following properties: 1. ; 2= F and X 2 F. 2. F is closed under forming finite intersections. 4 3. F is closed under taking supersets. If F is a filter on X, then the set of complements F c := fX n F j F 2 Fg is a generalized bornology on X. Vice versa the complements of a generalized bornology B is a filter if and only if X 62 B, i.e., B is not the maximal bornology. The generalized bornology F c is a bornology if and only if the filter is free, i.e., if and only T if F 2F F = ;. Hence, upon taking complements, non-maximal bornologies correspond to free filters on X. Example 1.13. Let F be a subset of PX . Then F ? := fY ⊆ X j (8L 2 F j jL \ Y j < 1g is a bornology. It is called the dual bornology to F. We have an obvious inclusion F ⊆ (F ?)?. Definition 1.14. A (generalized) bornological space is a pair (X; B) of a set with a (generalized) bornology. Usually we write X for generalized bornological spaces and let BX denote its generalized bornology. Example 1.15. For a set X we write Xmin and Xmax for X equipped with the minimal and maximal bornology. We write X; for the generalized bornological space with the empty bornology. Example 1.16. We consider a bornological space X and a subset L: Definition 1.17. L is called locally finite if jL \ Bj < 1 for all B in BX . We let LF(X) denote the set of locally finite subsets of X. By Example 1.13, the subset LF(X) of PX is a bornology on X. ? ? ? We write (X; LF(X)) =: X . With this notation we have (Xmin) = Xmax and (Xmax) = Xmin. Let f : X ! Y be a map between the underlying sets of generalized bornological spaces. Definition 1.18. f is called: −1 1. proper if f (BY ) ⊆ BX . 2. bornological, if f(BX ) ⊆ BY . 5 Let A be a subset of PY . By the following lemma we can check properness of a map on generators. Lemma 1.19. S 1.
Recommended publications
  • Linear Operators and Adjoints

    Linear Operators and Adjoints

    Chapter 6 Linear operators and adjoints Contents Introduction .......................................... ............. 6.1 Fundamentals ......................................... ............. 6.2 Spaces of bounded linear operators . ................... 6.3 Inverseoperators.................................... ................. 6.5 Linearityofinverses.................................... ............... 6.5 Banachinversetheorem ................................. ................ 6.5 Equivalenceofspaces ................................. ................. 6.5 Isomorphicspaces.................................... ................ 6.6 Isometricspaces...................................... ............... 6.6 Unitaryequivalence .................................... ............... 6.7 Adjoints in Hilbert spaces . .............. 6.11 Unitaryoperators ...................................... .............. 6.13 Relations between the four spaces . ................. 6.14 Duality relations for convex cones . ................. 6.15 Geometric interpretation of adjoints . ............... 6.15 Optimization in Hilbert spaces . .............. 6.16 Thenormalequations ................................... ............... 6.16 Thedualproblem ...................................... .............. 6.17 Pseudo-inverseoperators . .................. 6.18 AnalysisoftheDTFT ..................................... ............. 6.21 6.1 Introduction The field of optimization uses linear operators and their adjoints extensively. Example. differentiation, convolution, Fourier transform,
  • A Note on Riesz Spaces with Property-$ B$

    A Note on Riesz Spaces with Property-$ B$

    Czechoslovak Mathematical Journal Ş. Alpay; B. Altin; C. Tonyali A note on Riesz spaces with property-b Czechoslovak Mathematical Journal, Vol. 56 (2006), No. 2, 765–772 Persistent URL: http://dml.cz/dmlcz/128103 Terms of use: © Institute of Mathematics AS CR, 2006 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz Czechoslovak Mathematical Journal, 56 (131) (2006), 765–772 A NOTE ON RIESZ SPACES WITH PROPERTY-b S¸. Alpay, B. Altin and C. Tonyali, Ankara (Received February 6, 2004) Abstract. We study an order boundedness property in Riesz spaces and investigate Riesz spaces and Banach lattices enjoying this property. Keywords: Riesz spaces, Banach lattices, b-property MSC 2000 : 46B42, 46B28 1. Introduction and preliminaries All Riesz spaces considered in this note have separating order duals. Therefore we will not distinguish between a Riesz space E and its image in the order bidual E∼∼. In all undefined terminology concerning Riesz spaces we will adhere to [3]. The notions of a Riesz space with property-b and b-order boundedness of operators between Riesz spaces were introduced in [1]. Definition. Let E be a Riesz space. A set A E is called b-order bounded in ⊂ E if it is order bounded in E∼∼. A Riesz space E is said to have property-b if each subset A E which is order bounded in E∼∼ remains order bounded in E.
  • Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic

    Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic

    Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic Richard Blute University of Ottawa ongoing discussions with Geoff Crutwell, Thomas Ehrhard, Alex Hoffnung, Christine Tasson June 20, 2011 Richard Blute Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic Goals Develop a theory of (smooth) manifolds based on differential linear logic. Or perhaps develop a differential linear logic based on manifolds. Convenient vector spaces were recently shown to be a model. There is a well-developed theory of convenient manifolds, including infinite-dimensional manifolds. Convenient manifolds reveal additional structure not seen in finite dimensions. In particular, the notion of tangent space is much more complex. Synthetic differential geometry should also provide information. Convenient vector spaces embed into an extremely good model. Richard Blute Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic Convenient vector spaces (Fr¨olicher,Kriegl) Definition A vector space is locally convex if it is equipped with a topology such that each point has a neighborhood basis of convex sets, and addition and scalar multiplication are continuous. Locally convex spaces are the most well-behaved topological vector spaces, and most studied in functional analysis. Note that in any topological vector space, one can take limits and hence talk about derivatives of curves. A curve is smooth if it has derivatives of all orders. The analogue of Cauchy sequences in locally convex spaces are called Mackey-Cauchy sequences. The convergence of Mackey-Cauchy sequences implies the convergence of all Mackey-Cauchy nets. The following is taken from a long list of equivalences. Richard Blute Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic Convenient vector spaces II: Definition Theorem Let E be a locally convex vector space.
  • A Small Complete Category

    A Small Complete Category

    Annals of Pure and Applied Logic 40 (1988) 135-165 135 North-Holland A SMALL COMPLETE CATEGORY J.M.E. HYLAND Department of Pure Mathematics and Mathematical Statktics, 16 Mill Lane, Cambridge CB2 ISB, England Communicated by D. van Dalen Received 14 October 1987 0. Introduction This paper is concerned with a remarkable fact. The effective topos contains a small complete subcategory, essentially the familiar category of partial equiv- alence realtions. This is in contrast to the category of sets (indeed to all Grothendieck toposes) where any small complete category is equivalent to a (complete) poset. Note at once that the phrase ‘a small complete subcategory of a topos’ is misleading. It is not the subcategory but the internal (small) category which matters. Indeed for any ordinary subcategory of a topos there may be a number of internal categories with global sections equivalent to the given subcategory. The appropriate notion of subcategory is an indexed (or better fibred) one, see 0.1. Another point that needs attention is the definition of completeness (see 0.2). In my talk at the Church’s Thesis meeting, and in the first draft of this paper, I claimed too strong a form of completeness for the internal category. (The elementary oversight is described in 2.7.) Fortunately during the writing of [13] my collaborators Edmund Robinson and Giuseppe Rosolini noticed the mistake. Again one needs to pay careful attention to the ideas of indexed (or fibred) categories. The idea that small (sufficiently) complete categories in toposes might exist, and would provide the right setting in which to discuss models for strong polymorphism (quantification over types), was suggested to me by Eugenio Moggi.
  • Uniqueness of Von Neumann Bornology in Locally C∗-Algebras

    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.
  • Algebraic Cocompleteness and Finitary Functors

    Algebraic Cocompleteness and Finitary Functors

    Logical Methods in Computer Science Volume 17, Issue 2, 2021, pp. 23:1–23:30 Submitted Feb. 27, 2020 https://lmcs.episciences.org/ Published May 28, 2021 ALGEBRAIC COCOMPLETENESS AND FINITARY FUNCTORS JIRˇ´I ADAMEK´ ∗ Department of Mathematics, Czech Technical University in Prague, Czech Republic Institute of Theoretical Computer Science, Technical University Braunschweig, Germany e-mail address: [email protected] Abstract. A number of categories is presented that are algebraically complete and cocomplete, i.e., every endofunctor has an initial algebra and a terminal coalgebra. Example: assuming GCH, the category Set≤λ of sets of power at most λ has that property, whenever λ is an uncountable regular cardinal. For all finitary (and, more generally, all precontinuous) set functors the initial algebra and terminal coalgebra are proved to carry a canonical partial order with the same ideal completion. And they also both carry a canonical ultrametric with the same Cauchy completion. Finally, all endofunctors of the category Set≤λ are finitary if λ has countable cofinality and there are no measurable cardinals µ ≤ λ. 1. Introduction The importance of terminal coalgebras for an endofunctor F was clearly demonstrated by Rutten [Rut00]: for state-based systems whose state-objects lie in a category K and whose dynamics are described by F , the terminal coalgebra νF collects behaviors of individual states. And given a system A the unique coalgebra homomorphism from A to νF assigns to every state its behavior. However, not every endofunctor has a terminal coalgebra. Analogously, an initial algebra µF need not exist. Freyd [Fre70] introduced the concept of an algebraically complete category: this means that every endofunctor has an initial algebra.
  • On Bornivorous Set

    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.
  • Nakayama Categories and Groupoid Quantization

    Nakayama Categories and Groupoid Quantization

    NAKAYAMA CATEGORIES AND GROUPOID QUANTIZATION FABIO TROVA Abstract. We provide a precise description, albeit in the situation of standard categories, of the quantization functor Sum proposed by D.S. Freed, M.J. Hopkins, J. Lurie, and C. Teleman in a way enough abstract and flexible to suggest that an extension of the construction to the general context of (1; n)-categories should indeed be possible. Our method is in fact based primarily on dualizability and adjunction data, and is well suited for the homotopical setting. The construction also sheds light on the need of certain rescaling automorphisms, and in particular on the nature and properties of the Nakayama isomorphism. Contents 1. Introduction1 1.1. Conventions4 2. Preliminaries4 2.1. Duals4 2.2. Mates and Beck-Chevalley condition5 3. A comparison map between adjoints7 3.1. Projection formulas8 3.2. Pre-Nakayama map 10 3.3. Tensoring Kan extensions and pre-Nakayama maps 13 4. Nakayama Categories 20 4.1. Weights and the Nakayama map 22 4.2. Nakayama Categories 25 5. Quantization 26 5.1. A monoidal category of local systems 26 5.2. The quantization functors 28 5.3. Siamese twins 30 6. Linear representations of groupoids 31 References 33 arXiv:1602.01019v1 [math.CT] 2 Feb 2016 1. Introduction A k-extended n-dimensional C-valued topological quantum field theory n−k (TQFT for short) is a symmetric monoidal functor Z : Bordn !C, from the (1; k)-category of n-cobordisms extended down to codimension k [CS, Lu1] to a symmetric monoidal (1; k)-category C. For k = 1 one recovers the classical TQFT's as introduced in [At], while k = n gives rise to fully Date: October 22, 2018.
  • Baire Category Theorem and Its Consequences

    Baire Category Theorem and Its Consequences

    Functional Analysis, BSM, Spring 2012 Exercise sheet: Baire category theorem and its consequences S1 Baire category theorem. If (X; d) is a complete metric space and X = n=1 Fn for some closed sets Fn, then for at least one n the set Fn contains a ball, that is, 9x 2 X; r > 0 : Br(x) ⊂ Fn. Principle of uniform boundedness (Banach-Steinhaus). Let (X; k · kX ) be a Banach space, (Y; k · kY ) a normed space and Tn : X ! Y a bounded operator for each n 2 N. Suppose that for any x 2 X there exists Cx > 0 such that kTnxkY ≤ Cx for all n. Then there exists C > 0 such that kTnk ≤ C for all n. Inverse mapping theorem. Let X; Y be Banach spaces and T : X ! Y a bounded operator. Suppose that T is injective (ker T = f0g) and surjective (ran T = Y ). Then the inverse operator T −1 : Y ! X is necessarily bounded. 1. Consider the set Q of rational numbers with the metric d(x; y) = jx − yj; x; y 2 Q. Show that the Baire category theorem does not hold in this metric space. 2. Let (X; d) be a complete metric space. Suppose that Gn ⊂ X is a dense open set for n 2 N. Show that T1 n=1 Gn is non-empty. Find a non-complete metric space in which this is not true. T1 In fact, we can say more: n=1 Gn is dense. Prove this using the fact that a closed subset of a complete metric space is also complete.
  • Internal Groupoids and Exponentiability

    Internal Groupoids and Exponentiability

    VOLUME LX-4 (2019) INTERNAL GROUPOIDS AND EXPONENTIABILITY S.B. Niefield and D.A. Pronk Resum´ e.´ Nous etudions´ les objets et les morphismes exponentiables dans la 2-categorie´ Gpd(C) des groupoides internes a` une categorie´ C avec sommes finies lorsque C est : (1) finiment complete,` (2) cartesienne´ fermee´ et (3) localement cartesienne´ fermee.´ Parmi les exemples auxquels on s’interesse´ on trouve, en particulier, (1) les espaces topologiques, (2) les espaces com- pactement engendres,´ (3) les ensembles, respectivement. Nous considerons´ aussi les morphismes pseudo-exponentiables dans les categories´ “pseudo- slice” Gpd(C)==B. Comme ces dernieres` sont les categories´ de Kleisli d’une monade T sur la categorie´ “slice” stricte sur B, nous pouvons appliquer un theor´ eme` gen´ eral´ de Niefield [17] qui dit que si T Y est exponentiable dans une 2-categorie´ K, alors Y est pseudo-exponentiable dans la categorie´ de Kleisli KT . Par consequent,´ nous verrons que Gpd(C)==B est pseudo- cartesienne´ fermee,´ lorsque C est la categorie´ des espaces compactement en- gendres´ et chaque Bi est faiblement de Hausdorff, et Gpd(C) est localement pseudo-cartesienne´ fermee´ quand C est la categorie´ des ensembles ou une categorie´ localement cartesienne´ fermee´ quelconque. Abstract. We study exponentiable objects and morphisms in the 2-category Gpd(C) of internal groupoids in a category C with finite coproducts when C is: (1) finitely complete, (2) cartesian closed, and (3) locally cartesian closed. The examples of interest include (1) topological spaces, (2) com- pactly generated spaces, and (3) sets, respectively. We also consider pseudo- exponentiable morphisms in the pseudo-slice categories Gpd(C)==B.
  • Bounded Operator - Wikipedia, the Free Encyclopedia

    Bounded Operator - Wikipedia, the Free Encyclopedia

    Bounded operator - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Bounded_operator Bounded operator From Wikipedia, the free encyclopedia In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in X. In other words, there exists some M > 0 such that for all v in X The smallest such M is called the operator norm of L. A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless Y is the zero vector space. Rather, a bounded linear operator is a locally bounded function. A linear operator on a metrizable vector space is bounded if and only if it is continuous. Contents 1 Examples 2 Equivalence of boundedness and continuity 3 Linearity and boundedness 4 Further properties 5 Properties of the space of bounded linear operators 6 Topological vector spaces 7 See also 8 References Examples Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrix. Many integral transforms are bounded linear operators. For instance, if is a continuous function, then the operator defined on the space of continuous functions on endowed with the uniform norm and with values in the space with given by the formula is bounded.
  • Closed Graph Theorems for Bornological Spaces

    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.