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Uniform Boundedness Principle for Unbounded Operators
UNIFORM BOUNDEDNESS PRINCIPLE FOR UNBOUNDED OPERATORS C. GANESA MOORTHY and CT. RAMASAMY Abstract. A uniform boundedness principle for unbounded operators is derived. A particular case is: Suppose fTigi2I is a family of linear mappings of a Banach space X into a normed space Y such that fTix : i 2 Ig is bounded for each x 2 X; then there exists a dense subset A of the open unit ball in X such that fTix : i 2 I; x 2 Ag is bounded. A closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings as consequences of this principle. Some applications of this principle are also obtained. 1. Introduction There are many forms for uniform boundedness principle. There is no known evidence for this principle for unbounded operators which generalizes classical uniform boundedness principle for bounded operators. The second section presents a uniform boundedness principle for unbounded operators. An application to derive Hellinger-Toeplitz theorem is also obtained in this section. A JJ J I II closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings in the third section as consequences of this principle. Go back Let us assume the following: Every vector space X is over R or C. An α-seminorm (0 < α ≤ 1) is a mapping p: X ! [0; 1) such that p(x + y) ≤ p(x) + p(y), p(ax) ≤ jajαp(x) for all x; y 2 X Full Screen Close Received November 7, 2013. 2010 Mathematics Subject Classification. Primary 46A32, 47L60. Key words and phrases. -
Surjective Isometries on Grassmann Spaces 11
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of the Academy's Library SURJECTIVE ISOMETRIES ON GRASSMANN SPACES FERNANDA BOTELHO, JAMES JAMISON, AND LAJOS MOLNAR´ Abstract. Let H be a complex Hilbert space, n a given positive integer and let Pn(H) be the set of all projections on H with rank n. Under the condition dim H ≥ 4n, we describe the surjective isometries of Pn(H) with respect to the gap metric (the metric induced by the operator norm). 1. Introduction The study of isometries of function spaces or operator algebras is an important research area in functional analysis with history dating back to the 1930's. The most classical results in the area are the Banach-Stone theorem describing the surjective linear isometries between Banach spaces of complex- valued continuous functions on compact Hausdorff spaces and its noncommutative extension for general C∗-algebras due to Kadison. This has the surprising and remarkable consequence that the surjective linear isometries between C∗-algebras are closely related to algebra isomorphisms. More precisely, every such surjective linear isometry is a Jordan *-isomorphism followed by multiplication by a fixed unitary element. We point out that in the aforementioned fundamental results the underlying structures are algebras and the isometries are assumed to be linear. However, descriptions of isometries of non-linear structures also play important roles in several areas. We only mention one example which is closely related with the subject matter of this paper. This is the famous Wigner's theorem that describes the structure of quantum mechanical symmetry transformations and plays a fundamental role in the probabilistic aspects of quantum theory. -
Topological Vector Spaces and Algebras
Joseph Muscat 2015 1 Topological Vector Spaces and Algebras [email protected] 1 June 2016 1 Topological Vector Spaces over R or C Recall that a topological vector space is a vector space with a T0 topology such that addition and the field action are continuous. When the field is F := R or C, the field action is called scalar multiplication. Examples: A N • R , such as sequences R , with pointwise convergence. p • Sequence spaces ℓ (real or complex) with topology generated by Br = (a ): p a p < r , where p> 0. { n n | n| } p p p p • LebesgueP spaces L (A) with Br = f : A F, measurable, f < r (p> 0). { → | | } R p • Products and quotients by closed subspaces are again topological vector spaces. If π : Y X are linear maps, then the vector space Y with the ini- i → i tial topology is a topological vector space, which is T0 when the πi are collectively 1-1. The set of (continuous linear) morphisms is denoted by B(X, Y ). The mor- phisms B(X, F) are called ‘functionals’. +, , Finitely- Locally Bounded First ∗ → Generated Separable countable Top. Vec. Spaces ///// Lp 0 <p< 1 ℓp[0, 1] (ℓp)N (ℓp)R p ∞ N n R 2 Locally Convex ///// L p > 1 L R , C(R ) R pointwise, ℓweak Inner Product ///// L2 ℓ2[0, 1] ///// ///// Locally Compact Rn ///// ///// ///// ///// 1. A set is balanced when λ 6 1 λA A. | | ⇒ ⊆ (a) The image and pre-image of balanced sets are balanced. ◦ (b) The closure and interior are again balanced (if A 0; since λA = (λA)◦ A◦); as are the union, intersection, sum,∈ scaling, T and prod- uct A ⊆B of balanced sets. -
Bornologically Isomorphic Representations of Tensor Distributions
Bornologically isomorphic representations of distributions on manifolds E. Nigsch Thursday 15th November, 2018 Abstract Distributional tensor fields can be regarded as multilinear mappings with distributional values or as (classical) tensor fields with distribu- tional coefficients. We show that the corresponding isomorphisms hold also in the bornological setting. 1 Introduction ′ ′ ′r s ′ Let D (M) := Γc(M, Vol(M)) and Ds (M) := Γc(M, Tr(M) ⊗ Vol(M)) be the strong duals of the space of compactly supported sections of the volume s bundle Vol(M) and of its tensor product with the tensor bundle Tr(M) over a manifold; these are the spaces of scalar and tensor distributions on M as defined in [?, ?]. A property of the space of tensor distributions which is fundamental in distributional geometry is given by the C∞(M)-module isomorphisms ′r ∼ s ′ ∼ r ′ Ds (M) = LC∞(M)(Tr (M), D (M)) = Ts (M) ⊗C∞(M) D (M) (1) (cf. [?, Theorem 3.1.12 and Corollary 3.1.15]) where C∞(M) is the space of smooth functions on M. In[?] a space of Colombeau-type nonlinear generalized tensor fields was constructed. This involved handling smooth functions (in the sense of convenient calculus as developed in [?]) in par- arXiv:1105.1642v1 [math.FA] 9 May 2011 ∞ r ′ ticular on the C (M)-module tensor products Ts (M) ⊗C∞(M) D (M) and Γ(E) ⊗C∞(M) Γ(F ), where Γ(E) denotes the space of smooth sections of a vector bundle E over M. In[?], however, only minor attention was paid to questions of topology on these tensor products. -
An Additivity Theorem for Uniformly Continuous Functions
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 146–147 (2005) 339–352 www.elsevier.com/locate/topol An additivity theorem for uniformly continuous functions Alessandro Berarducci a,1, Dikran Dikranjan b,∗,1, Jan Pelant c a Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, 56127 Pisa, Italy b Dipartimento di Matematica e Informatica, Università di Udine, Via Delle Scienze 206, 33100 Udine, Italy c Institute of Mathematics, Czech Academy of Sciences, Žitna 25 11567 Praha 1, Czech Republic Received 6 November 2002; received in revised form 21 May 2003; accepted 30 May 2003 Abstract We consider metric spaces X with the nice property that any continuous function f : X → R which is uniformly continuous on each set of a finite cover of X by closed sets, is itself uniformly continuous. We characterize the spaces with this property within the ample class of all locally connected metric spaces. It turns out that they coincide with the uniformly locally connected spaces, so they include, for instance, all topological vector spaces. On the other hand, in the class of all totally disconnected spaces, these spaces coincide with the UC spaces. 2004 Elsevier B.V. All rights reserved. MSC: primary 54E15, 54E40; secondary 54D05, 54E50 Keywords: Uniform continuity; Metric spaces; Local connectedness; Total disconnectedness; Atsuji space 1. Introduction All spaces in the sequel are metric and C(X) denotes the set of all continuous real- valued functions of a space X. The main goal of this paper is to study the following natural notion: * Corresponding author. -
Functional Analysis Solutions to Exercise Sheet 6
Institut für Analysis WS2019/20 Prof. Dr. Dorothee Frey 21.11.2019 M.Sc. Bas Nieraeth Functional Analysis Solutions to exercise sheet 6 Exercise 1: Properties of compact operators. (a) Let X and Y be normed spaces and let T 2 K(X; Y ). Show that R(T ) is separable. (b) Let X, Y , and Z be Banach spaces. Let T 2 K(X; Y ) and S 2 L(Y; Z), where S is injective. Prove that for every " > 0 there is a constant C ≥ 0 such that kT xk ≤ "kxk + CkST xk for all x 2 X. Solution: (a) Since bounded sets have a relatively compact image under T , for each n 2 N the sets T (nBX ) are totally bounded, where nBX := fx 2 X : kxk ≤ ng. Then it follows from Exercise 1 of Exercise sheet 3 that T (nBX ) is separable. But then ! [ [ T (nBX ) = T nBX = T (X) = R(T ) n2N n2N is also separable, as it is a countable union of separable sets. The assertion follows. (b) First note that by using a homogeneity argument, one finds that this result is true, ifand only it is true for x 2 X with kxk = 1. Thus, it remains to prove the result for x 2 X with kxk = 1. We argue by contradiction. If the result does not hold, then there is an " > 0 such that for all C ≥ 0 there exists and x 2 X with kxk = 1 such that kT xk > " + CkST xk. By choosing C = n for n 2 N, we find a sequence (xn)n2N in X with kxnk = 1 such that kT xnk > " + nkST xnk: (1) Since (xn)n2N is bounded and T is compact, there exists a subsequence (T xnj )j2N with limit y 2 Y . -
S-Barrelled Topological Vector Spaces*
Canad. Math. Bull. Vol. 21 (2), 1978 S-BARRELLED TOPOLOGICAL VECTOR SPACES* BY RAY F. SNIPES N. Bourbaki [1] was the first to introduce the class of locally convex topological vector spaces called "espaces tonnelés" or "barrelled spaces." These spaces have some of the important properties of Banach spaces and Fréchet spaces. Indeed, a generalized Banach-Steinhaus theorem is valid for them, although barrelled spaces are not necessarily metrizable. Extensive accounts of the properties of barrelled locally convex topological vector spaces are found in [5] and [8]. In the last few years, many new classes of locally convex spaces having various types of barrelledness properties have been defined (see [6] and [7]). In this paper, using simple notions of sequential convergence, we introduce several new classes of topological vector spaces which are similar to Bourbaki's barrelled spaces. The most important of these we call S-barrelled topological vector spaces. A topological vector space (X, 2P) is S-barrelled if every sequen tially closed, convex, balanced, and absorbing subset of X is a sequential neighborhood of the zero vector 0 in X Examples are given of spaces which are S-barrelled but not barrelled. Then some of the properties of S-barrelled spaces are given—including permanence properties, and a form of the Banach- Steinhaus theorem valid for them. S-barrelled spaces are useful in the study of sequentially continuous linear transformations and sequentially continuous bilinear functions. For the most part, we use the notations and definitions of [5]. Proofs which closely follow the standard patterns are omitted. 1. Definitions and Examples. -
Extension of Compact Operators from DF-Spaces to C(K) Spaces
Applied General Topology c Universidad Polit´ecnica de Valencia @ Volume 7, No. 2, 2006 pp. 165-170 Extension of Compact Operators from DF-spaces to C(K) spaces Fernando Garibay Bonales and Rigoberto Vera Mendoza Abstract. It is proved that every compact operator from a DF- space, closed subspace of another DF-space, into the space C(K) of continuous functions on a compact Hausdorff space K can be extended to a compact operator of the total DF-space. 2000 AMS Classification: Primary 46A04, 46A20; Secondary 46B25. Keywords: Topological vector spaces, DF-spaces and C(K) the spaces. 1. Introduction Let E and X be topological vector spaces with E a closed subspace of X. We are interested in finding out when a continuous operator T : E → C(K) has an extension T˜ : X → C(K), where C(K) is the space of continuous real functions on a compact Hausdorff space K and C(K) has the norm of the supremum. When this is the case we will say that (E,X) has the extension property. Several advances have been made in this direction, a basic resume and bibliography for this problem can be found in [5]. In this work we will focus in the case when the operator T is a compact operator. In [4], p.23 , it is proved that (E,X) has the extension property when E and X are Banach spaces and T : E → C(K) is a compact operator. In this paper we extend this result to the case when E and X are DF-spaces (to be defined below), for this, we use basic tools from topological vector spaces. -
The Banach-Alaoglu Theorem for Topological Vector Spaces
The Banach-Alaoglu theorem for topological vector spaces Christiaan van den Brink a thesis submitted to the Department of Mathematics at Utrecht University in partial fulfillment of the requirements for the degree of Bachelor in Mathematics Supervisor: Fabian Ziltener date of submission 06-06-2019 Abstract In this thesis we generalize the Banach-Alaoglu theorem to topological vector spaces. the theorem then states that the polar, which lies in the dual space, of a neighbourhood around zero is weak* compact. We give motivation for the non-triviality of this theorem in this more general case. Later on, we show that the polar is sequentially compact if the space is separable. If our space is normed, then we show that the polar of the unit ball is the closed unit ball in the dual space. Finally, we introduce the notion of nets and we use these to prove the main theorem. i ii Acknowledgments A huge thanks goes out to my supervisor Fabian Ziltener for guiding me through the process of writing a bachelor thesis. I would also like to thank my girlfriend, family and my pet who have supported me all the way. iii iv Contents 1 Introduction 1 1.1 Motivation and main result . .1 1.2 Remarks and related works . .2 1.3 Organization of this thesis . .2 2 Introduction to Topological vector spaces 4 2.1 Topological vector spaces . .4 2.1.1 Definition of topological vector space . .4 2.1.2 The topology of a TVS . .6 2.2 Dual spaces . .9 2.2.1 Continuous functionals . -
ARZEL`A-ASCOLI's THEOREM in UNIFORM SPACES Mateusz Krukowski 1. Introduction. Around 1883, Cesare Arzel`
DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018020 DYNAMICAL SYSTEMS SERIES B Volume 23, Number 1, January 2018 pp. 283{294 ARZELA-ASCOLI'S` THEOREM IN UNIFORM SPACES Mateusz Krukowski∗ L´od´zUniversity of Technology, Institute of Mathematics W´olcza´nska 215, 90-924L´od´z, Poland Abstract. In the paper, we generalize the Arzel`a-Ascoli'stheorem in the setting of uniform spaces. At first, we recall the Arzel`a-Ascolitheorem for functions with locally compact domains and images in uniform spaces, coming from monographs of Kelley and Willard. The main part of the paper intro- duces the notion of the extension property which, similarly as equicontinuity, equates different topologies on C(X; Y ). This property enables us to prove the Arzel`a-Ascoli'stheorem for uniform convergence. The paper culminates with applications, which are motivated by Schwartz's distribution theory. Using the Banach-Alaoglu-Bourbaki's theorem, we establish the relative compactness of 0 n subfamily of C(R; D (R )). 1. Introduction. Around 1883, Cesare Arzel`aand Giulio Ascoli provided the nec- essary and sufficient conditions under which every sequence of a given family of real-valued continuous functions, defined on a closed and bounded interval, has a uniformly convergent subsequence. Since then, numerous generalizations of this result have been obtained. For instance, in [10], p. 278 the compact families of C(X; R), with X a compact space, are exactly those which are equibounded and equicontinuous. The space C(X; R) is given with the standard norm kfk := sup jf(x)j; x2X where f 2 C(X; R). -
Ark3: Exercises for MAT2400 — Spaces of Continuous Functions
Ark3: Exercises for MAT2400 — Spaces of continuous functions The exercises on this sheet covers the sections 3.1 to 3.4 in Tom’s notes. They are to- gether with a few problems from Ark2 the topics for the groups on Thursday, February 16 and Friday, February 17. With the following distribution: Thursday, February 16: No 3, 4, 5, 6, 9, 10, 12. From Ark2: No 27, 28 Friday, February 17: No: 1,2, 7, 8, 11, 13 Key words: Uniform continuity, equicontinuity, uniform convergence, spaces of conti- nuous functions. Uniform continuity and equicontinuity Problem 1. ( Tom’s notes 1, Problem 3.1 (page 50)). Show that x2 is not uniformly continuous on R. Hint: x2 y2 =(x + y)(x y). − − Problem 2. ( Tom’s notes 2, Problem 3.1 (page 51)). Show that f :(0, 1) R given 1 → by f(x)= x is not uniformly continuous. Problem 3. a) Let I be an interval and assume that f is a function differentiable in I. Show that if the derivative f is bounded in I, then f is uniformly continuous in I. Hint: Use the mean value theorem. b) Show that the function √x is uniformly continuous on [1, ). Is it uniformly con- tinuous on [0, )? ∞ ∞ Problem 4. Let F C([a, b]) denote a subset whose members all are differentiable ⊆ on (a, b). Assume that there is a constant M such that f (x) M for all x (a, b) | |≤ ∈ and all f . Show that F is equicontinuous. Hint: Use the mean value theorem. ∈F Problem 5. Let Pn be the subset of C([0, 1]) whose elements are the poly’s of degree at most n.LetM be a positive constant and let S Pn be the subset of poly’s with coefficients bounded by M; i.e., the set of poly’s ⊆n a T i with a M. -
The Arzel`A-Ascoli Theorem
John Nachbar Washington University March 27, 2016 The Arzel`a-AscoliTheorem The Arzel`a-AscoliTheorem gives sufficient conditions for compactness in certain function spaces. Among other things, it helps provide some additional perspective on what compactness means. Let C([0; 1]) denote the set of continuous functions f : [0; 1] ! R. Because the domain is compact, one can show (I leave this as an exercise) that any f 2 C([0; 1]) is uniformly continuous: for any " > 0 there is a δ > 0 such that if jx − x^j < δ then jf(x) − f(^x)j < ". An example of a function that is continuous but not uniformly continuous is f : (0; 1] ! R given by f(x) = 1=x. A set F ⊆ C([0; 1]) is (uniformly) equicontinuous iff for any " > 0 there exists a δ > 0 such that for all f 2 F, if jx − x^j < δ then jf(x) − f(^x)j < ". That is, if F is (uniformly) equicontinuous then every f 2 F is uniformly continuous and for every " > 0 and every f 2 F, I can use the same δ > 0. Example3 below gives an example of a set of (uniformly) continuous functions that is not equicontinuous. A trivial example of an equicontinuous set of functions is a set of functions such that any pair of functions differ from each other by an additive constant: for any f and g in the set, there is an a such that for all x 2 [0; 1], f(x) = f^(x) + a. A more interesting example is given by a set of differentiable functions for which the derivative is uniformly bounded: there is a W > 0 such that for all x 2 [0; 1] and all , jDf(x)j < W .