DOCSLIB.ORG

Nuclear Mappings and Quasi-Nuclear Mappings

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Journal of Information and Computational Science ISSN: 1548-7741 NUCLEAR MAPPINGS AND QUASI-NUCLEAR MAPPINGS Dr. Santosh Kumar1, Dr Raj Kumar2 1Faculty, Department of Mathematics, Patna Science College, Patna University, Patna, Bihar. E-Mail: [email protected] 2St Xavier’s College of Management and Technology, Digha Ashiyana Road, Patna, Bihar. E-Mail: [email protected] ABSTRACT In the present paper, we look at different approaches towards tensor products, which are known to coincide for Banach spaces, and give necessary and sufficient conditions for the dual spaces of the particular semi Banach spaces. A while later we consider the injective tensor standard as one especially well known model for standards on tensor results of semi Banach spaces. As we will see, there are three different descriptions of injective quasi-norms. This condition turns out to be theinsignificant necessity. Additionally, we study augmentations of the old style injective and p-nuclear tensor standards to semi Banach spaces. Specifically, we give an adequate condition for the p-atomic semi standards to be crossnorms, which especially applies to the case of weighted 푙푝-sequence spaces. KEYWORDS- Algebraic Tensor Product, Quasi-Norm, P-Norm, Injective Tensor Norm, P- Nuclear Tensor Norm. 1. INTRODUCTION In mathematics, a nuclear space is a topological vector space with various better properties of finite-dimensional vector spaces. The topology on them can be characterized by a family of seminorms whose unit balls decrease quickly in size. Vector spaces whose components are "smooth" in some sense will in general be atomic spaces; a run of the mill case of an atomic space is the arrangement of smooth capacities on a conservative manifold [1]. All limited dimensional vector spaces are nuclear (in light of the fact that each administrator on a limited dimensional vector space is nuclear). There are no Banach spaces that are nuclear, Volume 10 Issue 7 - 2020 390 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741 with the exception of the limited dimensional ones. By and by a kind of banter to this is regularly valid: in the event that a "normally happening" topological vector space isn't a Banach space, at that point there is a decent possibility that it is nuclear. While for tensor results of Hilbert spaces, a significant number of these perspectives are surely known because of the unique properties of these Hilbert spaces, the hypothesis of tensor results of Banach spaces is unmistakably increasingly included. Be that as it may, in present day estimate hypothesis semi Banach spaces turned out to be progressively significant. In this setting they regularly show up while portraying purported estimate spaces, for example given a fixed estimation strategy one is keen on the assortment of all capacities with a given combination rate[2]. Definition 1: A nuclear space is a nearby convex topological vector space for instances any seminorm p we can get a larger seminorm q so as to the normal chart from Vq to Vp is nuclear. Casually, this implies at whatever point we are given the unit wad of some seminorm, we can locate an "a lot littler" unit bundle of another seminorm inside it, or that any area of 0 contains an "a lot littler" neighborhood. It isn't important to check this condition for all seminorms p; it is adequate to check it for a lot of seminorms that produce the topology, as it were, a lot of seminorms that are a sub base for the topology. Rather than utilizing subjective Banach spaces and nuclear administrators, we can give a definition as far as Hilbert spaces and follow class administrators, which are more obvious. (On Hilbert spaces nuclear administrators are frequently called follow class administrators.)We will state that seminorm p is a Hilbert seminorm if Vp is a Hilbert space, or equivalently if p comes from a sesquilinear optimistic semi definite form on V. Definition 2: A nuclear space is a topological vector space through topology definite by a family of Hilbert seminorms, such that for some Hilbert seminorm p we can discover a better Hilbert seminorm q so that the ordinary chart from Vq to Vp is trace class. Some authors prefer to use Hilbert–Schmidt operators rather than follow class administrators. This has little effect, in light of the fact that any follow class administrator is Hilbert–Schmidt, and the result of two Hilbert–Schmidt administrators is of follow class. Definition 3: An nuclear space is a topological vector space with a topology characterized by a group of Hilbert seminorms, to such an extent that for any Hilbert seminorm p we can find a larger Hilbert seminorm q so that the natural map from Vq to Vp is Hilbert–Schmidt. On the off chance that we are happy to utilize the idea of an nuclear administrator from a self- assertive locally raised topological vector space to a Banach space, we can give shorter definitions as pursues: Definition 4: A nuclear space is a locally convex topological vector space such that for any seminorm p the natural map from V to Vp is nuclear. Volume 10 Issue 7 - 2020 391 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741 Definition 5: A nuclear space is a locally arched topological vector space with the end goal that any persistent straight guide to a Banach space is nuclear. Grothendieck utilized a definition like the accompanying one: Definition 6: A nuclear space in the vicinity arched topological vector space A with the end goal that for any locally curved topological vector space B the characteristic guide from the projective to the injective tensor item of A and B is an isomorphism [3]. 2. TENSOR PRODUCTS OF (QUASI-)BANACH SPACE 2.1. ALGEBRAIC AND ANALYTIC DEFINITION In algebra tensor product constructions are known for several different structures. The starting point for one possibility of an explicit construction for vector spaces X and Y (with respect to the same field; Thus we focus on real or complex vector spaces) is the free vector space F(X,Y) on X × Y , i.e. the set Afterwards the algebraic tensor product X ⊗ Y is defined as the quotient space of F(X, Y) with respect to the subspace In this way the canonical mapping (x, y)→ x ⊗ y from X × Y to X ⊗ Y becomes bilinear. The common practical explanatory approach for normed spaces X and Y is marginally different. Once more one starts with F(X, Y), but this times this space is equipped with the following 푛 ′ equivalence relation. We say 푓 = ∑푗=1 휆푖푥푗⨂푦푖 ∈ 퐹(푋, 푌) generates an operator 퐴푓: 푋 → Y by the determination i.e. f and g generate the same operator from the dual space 푋′of X to Y . Of interest now is the quotient space 푋⨂퐴푌 = F(X, Y )/ +, which is found to coincide as a vector space with X ⊗ Y Volume 10 Issue 7 - 2020 392 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741 By this definition the connection with linear mappings from 푋′to Y is made obvious right from the beginning. This systematic approach applies to semi normed spaces also, however since the double space is potentially minor, this identicalness connection just as the separate remainder space may get inconsequential. To stay away from this, for example to guarantee the proportionality of the two methodologies, we need to force certain confinements on the semi normed spaces. This circumstance is explained by the accompanying hypothesis. Theorem 1. Let X and Y be two quasi-normed spaces. Then it holds X ⊗ Y = X ⊗A Y if, and ′ only if, 푋 Separates the points in X, i.e. for every x ∈ X \ {0} there exists a functional 휑푥 ∈ ′ 푋 , such that 휑푥(x) = 0. A quasi-Banach space X with this property is said to have a separating dual. Proof. So as to show the occurrence of the two spaces we need to show that U = V= {f∈F(X, Y): 퐴푓 = 0 } grasp. The insertion U ⊂ V is apparent. For the repealen closure we comment that the situation on 푋′ is corresponding to Ax⊗y = 0 for every x = 0 and y = 0. To show now V ⊂ U, we show instead, that from f∈ U follows f ∈ V. We shall utilize the fact that for each f 푛 ∈ U there survive an (algebraically) corresponding illustration푓 = ∑푖=1 푥푖 ⨂ 푦푖, where {푥1, . 1 푛 . , 푥푛} ⊂ X and {푦 , . , 푦 } ⊂ Y are linearly autonomous (this can be seen analogously to 1 푛 [Lemma 1.1]). The linearity of f) →퐴푓, the linear independency of {푦 , . , 푦 } and the ′ assumption for 푋 (applied to the vectors 푥푖 = 0 now yield 퐴푓 = 0. In a similar way, we can also consider operators. We then observe that Holds for all f, g ∈ X ⊗ Y if, and only if, 푋′ and 푌′are separating. This follows from F(X, Y) ≅ F(Y, X) and 푋⨂퐴푌 ≅ X ⊗ Y ≅ Y ⊗ X ≅ Y ⨂퐴푋 , where the isomorphism is provided by the canonical identification X ⊗ Y)→ Y ⊗ X, x ∈ X, y ∈ Y (for the logarithmic tensor item this is in every case genuine, and the suspicions guarantee this reaches out to the separate (useful diagnostic) comparability relations). Because of this perception we hereafter consistently expect that X and Y both have isolating duals (without in every case unequivocally referencing it). REMARK1. If one is merely involved in equipping the algebraic tensor product X ⊗ Y of general topological vector spaces with just some topological structure, then one does not need information on the respective dual spaces.
Recommended publications
  • Projective Tensor Products, Injective Tensor Products, and Dual Relations on Operator Spaces

    Projective Tensor Products, Injective Tensor Products, and Dual Relations on Operator Spaces

    University of Windsor Scholarship at UWindsor Electronic Theses and Dissertations Theses, Dissertations, and Major Papers 1-1-2006 Projective tensor products, injective tensor products, and dual relations on operator spaces. Fuhua Chen University of Windsor Follow this and additional works at: https://scholar.uwindsor.ca/etd Recommended Citation Chen, Fuhua, "Projective tensor products, injective tensor products, and dual relations on operator spaces." (2006). Electronic Theses and Dissertations. 7096. https://scholar.uwindsor.ca/etd/7096 This online database contains the full-text of PhD dissertations and Masters’ theses of University of Windsor students from 1954 forward. These documents are made available for personal study and research purposes only, in accordance with the Canadian Copyright Act and the Creative Commons license—CC BY-NC-ND (Attribution, Non-Commercial, No Derivative Works). Under this license, works must always be attributed to the copyright holder (original author), cannot be used for any commercial purposes, and may not be altered. Any other use would require the permission of the copyright holder. Students may inquire about withdrawing their dissertation and/or thesis from this database. For additional inquiries, please contact the repository administrator via email ([email protected]) or by telephone at 519-253-3000ext. 3208. P r o je c t iv e T e n so r P r o d u c t s , In je c t iv e T e n so r P r o d u c t s , a n d D ual R elatio ns on O p e r a t o r S paces by Fuhua Chen A Thesis Submitted to the Faculty of Graduate Studies and Research through Mathematics and Statistics in Partial Fulfillment of the Requirements of the Degree of Master of Science at the University of Windsor Windsor, Ontario, Canada 2006 © 2006 Fuhua Chen Reproduced with permission of the copyright owner.
  • Grothendieck's Tensor Norms

    Grothendieck's Tensor Norms

    Grothendieck's tensor norms Vandana Shivaji College, University of Delhi, India April, 2015 Vandana Grothendieck's tensor norms Overview Grothendieck's tensor norms Operator spaces Tensor products of Operator spaces Schur tensor product Vandana Grothendieck's tensor norms Grothendieck's tensor norms A Banach space is a complete normed space. For Banach spaces X and Y , X ⊗ Y = spanfx ⊗ y : x 2 X; y 2 Y g, where x ⊗ y is the functional on B(X∗ × Y ∗; C) given by x ⊗ y(f; g) = f(x)g(y) for f 2 X∗ and g 2 Y ∗. For a pair of arbitrary Banach spaces X and Y , the norm on X ⊗ Y induced by the embedding X ⊗ Y ! B(X∗ × Y ∗; C) is known as Banach space injective tensor norm. That is , for u 2 X ⊗ Y , the Banach space injective tensor norm is defined to be n n o X ∗ ∗ kukλ = sup f(xi)g(yi) : f 2 X1 ; g 2 Y1 : i=1 Vandana Grothendieck's tensor norms Grothendieck's tensor norms Question is How can we norm on X ⊗ Y ? n X kx ⊗ ykα ≤ kxkkyk, then, for u = xi ⊗ yi, by triangle's i=1 n X inequality it follows that kukα ≤ kxikkyik. Since this holds for i=1 n X every representation of u, so we have kukα ≤ inff kxikkyikg. i=1 For a pair of arbitrary Banach spaces X and Y and u an element in the algebraic tensor product X ⊗ Y , the Banach space projective tensor norm is defined to be n n X X kukγ = inff kxikkyik : u = xi ⊗ yi; n 2 Ng: i=1 i=1 X ⊗γ Y will denote the completion of X ⊗ Y with respect to this norm.
  • Uniform Boundedness Principle for Unbounded Operators

    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.
  • Distinguished Property in Tensor Products and Weak* Dual Spaces

    Distinguished Property in Tensor Products and Weak* Dual Spaces

    axioms Article Distinguished Property in Tensor Products and Weak* Dual Spaces Salvador López-Alfonso 1 , Manuel López-Pellicer 2,* and Santiago Moll-López 3 1 Department of Architectural Constructions, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] 2 Emeritus and IUMPA, Universitat Politècnica de València, 46022 Valencia, Spain 3 Department of Applied Mathematics, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] * Correspondence: [email protected] 0 Abstract: A local convex space E is said to be distinguished if its strong dual Eb has the topology 0 0 0 0 b(E , (Eb) ), i.e., if Eb is barrelled. The distinguished property of the local convex space Cp(X) of real- valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and Cp-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space Cp(X) is distinguished if and only if any function f 2 RX belongs to the pointwise closure of a pointwise bounded set in C(X). The extensively studied distinguished properties in the injective tensor products Cp(X) ⊗# E and in Cp(X, E) contrasts with the few distinguished properties of injective tensor products related to the dual space Lp(X) of Cp(X) endowed with the weak* topology, as well as to the weak* dual of Cp(X, E). To partially fill this gap, some distinguished properties in the injective tensor product space Lp(X) ⊗# E are presented and a characterization of the distinguished property of the weak* dual of Cp(X, E) for wide classes of spaces X and E is provided.
  • Topological Vector Spaces and Algebras

    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.
  • On the Geometry of Projective Tensor Products

    On the Geometry of Projective Tensor Products

    ON THE GEOMETRY OF PROJECTIVE TENSOR PRODUCTS OHAD GILADI, JOSCHA PROCHNO, CARSTEN SCHUTT,¨ NICOLE TOMCZAK-JAEGERMANN, AND ELISABETH WERNER n Abstract. In this work, we study the volume ratio of the projective tensor products `p ⊗π n n `q ⊗π `r with 1 ≤ p ≤ q ≤ r ≤ 1. We obtain asymptotic formulas that are sharp in almost all cases. As a consequence of our estimates, these spaces allow for a nearly Euclidean decomposition of Kaˇsintype whenever 1 ≤ p ≤ q ≤ r ≤ 2 or 1 ≤ p ≤ 2 ≤ r ≤ 1 and q = 2. Also, from the Bourgain-Milman bound on the volume ratio of Banach spaces in terms of their cotype 2 constant, we obtain information on the cotype of these 3-fold projective tensor products. Our results naturally generalize to k-fold products `n ⊗ · · · ⊗ `n with k 2 p1 π π pk N and 1 ≤ p1 ≤ · · · ≤ pk ≤ 1. 1. Introduction In the geometry of Banach spaces the volume ratio vr(X) of an n-dimensional normed space X is defined as the n-th root of the volume of the unit ball in X divided by the volume of its John ellipsoid. This notion plays an important role in the local theory of Banach spaces and has significant applications in approximation theory. It formally originates in the works [Sza78] and [STJ80], which were influenced by the famous paper of B. Kaˇsin[Kaˇs77] on nearly Euclidean orthogonal decompositions. Kaˇsindiscovered that for arbitrary n 2 N, 2n the space `1 contains two orthogonal subspaces which are nearly Euclidean, meaning that n their Banach-Mazur distance to `2 is bounded by an absolute constant.
  • Bornologically Isomorphic Representations of Tensor Distributions

    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.
  • Tensor Products of Convex Cones, Part I: Mapping Properties, Faces, and Semisimplicity

    Tensor Products of Convex Cones, Part I: Mapping Properties, Faces, and Semisimplicity

    Tensor Products of Convex Cones, Part I: Mapping Properties, Faces, and Semisimplicity Josse van Dobben de Bruyn 24 September 2020 Abstract The tensor product of two ordered vector spaces can be ordered in more than one way, just as the tensor product of normed spaces can be normed in multiple ways. Two natural orderings have received considerable attention in the past, namely the ones given by the projective and injective (or biprojective) cones. This paper aims to show that these two cones behave similarly to their normed counterparts, and furthermore extends the study of these two cones from the algebraic tensor product to completed locally convex tensor products. The main results in this paper are the following: (i) drawing parallels with the normed theory, we show that the projective/injective cone has mapping properties analogous to those of the projective/injective norm; (ii) we establish direct formulas for the lineality space of the projective/injective cone, in particular providing necessary and sufficient conditions for the cone to be proper; (iii) we show how to construct faces of the projective/injective cone from faces of the base cones, in particular providing a complete characterization of the extremal rays of the projective cone; (iv) we prove that the projective/injective tensor product of two closed proper cones is contained in a closed proper cone (at least in the algebraic tensor product). 1 Introduction 1.1 Outline Tensor products of ordered (topological) vector spaces have been receiving attention for more than 50 years ([Mer64], [HF68], [Pop68], [PS69], [Pop69], [DS70], [vGK10], [Wor19]), but the focus has mostly been on Riesz spaces ([Sch72], [Fre72], [Fre74], [Wit74], [Sch74, §IV.7], [Bir76], [FT79], [Nie82], [GL88], [Nie88], [Bla16]) or on finite-dimensional spaces ([BL75], [Bar76], [Bar78a], [Bar78b], [Bar81], [BLP87], [ST90], [Tam92], [Mul97], [Hil08], [HN18], [ALPP19]).
  • 1 Lecture 09

    1 Lecture 09

    Notes for Functional Analysis Wang Zuoqin (typed by Xiyu Zhai) September 29, 2015 1 Lecture 09 1.1 Equicontinuity First let's recall the conception of \equicontinuity" for family of functions that we learned in classical analysis: A family of continuous functions defined on a region Ω, Λ = ffαg ⊂ C(Ω); is called an equicontinuous family if 8 > 0; 9δ > 0 such that for any x1; x2 2 Ω with jx1 − x2j < δ; and for any fα 2 Λ, we have jfα(x1) − fα(x2)j < . This conception of equicontinuity can be easily generalized to maps between topological vector spaces. For simplicity we only consider linear maps between topological vector space , in which case the continuity (and in fact the uniform continuity) at an arbitrary point is reduced to the continuity at 0. Definition 1.1. Let X; Y be topological vector spaces, and Λ be a family of continuous linear operators. We say Λ is an equicontinuous family if for any neighborhood V of 0 in Y , there is a neighborhood U of 0 in X, such that L(U) ⊂ V for all L 2 Λ. Remark 1.2. Equivalently, this means if x1; x2 2 X and x1 − x2 2 U, then L(x1) − L(x2) 2 V for all L 2 Λ. Remark 1.3. If Λ = fLg contains only one continuous linear operator, then it is always equicontinuous. 1 If Λ is an equicontinuous family, then each L 2 Λ is continuous and thus bounded. In fact, this boundedness is uniform: Proposition 1.4. Let X,Y be topological vector spaces, and Λ be an equicontinuous family of linear operators from X to Y .
  • S-Barrelled Topological Vector Spaces*

    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

    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.
  • Geometry in Tensor Products

    Geometry in Tensor Products

    Universita` degli Studi di Napoli \Federico II" Facolt`adi Scienze Matematiche Fisiche e Naturali Dipartimento di Matematica e Applicazioni \R.Caccioppoli" Dottorato di Ricerca in Scienze Matematiche Daniele Puglisi Geometry in tensor products TESI DI DOTTORATO XIX CICLO 2003-2007 Tutors: Prof. Paolo De Lucia Coordinatore: Prof. Salvatore Rionero Prof. Joseph Diestel ii Acknowledgements First of all I would like to thank Prof. Joseph Diestel, from Kent State Uni- versity, for his precious suggestions and many many helps. To sit near Joe when he is talking about math in his office, in some bar or also inside some airport, is an experience every young mathematicians should have. I really want to thank him to invited me and for giving me a comfortable permanence in Kent (I have no words to thank you!). Then I thank my advisor Prof. Paolo De Lucia for accepting me as his stu- dent and for giving me the opportunity to visit the Kent State University. Thanks to Prof. Richard Aron for his excellent mathematical support, for helping me in many situation, and for any time that he took me to hear the Cleveland Orchestra. I also want to dedicate part of my dissertation to Roberto Lo Re, Giovanni Cutolo, and Chansung Choi, they influenced strongly my life. Thanks to Prof. Artem Zvavitch for his support, and thanks to Giuseppe Saluzzo, Assunta Tataranni for the many helps. I want to give thanks the secretary support to the Department of Mathemat- ical Sciences at at Kent State University and at Napoli. Specially, I wish to thank Virginia Wright, Misty Tackett, Luciana Colmayer and Luisa Falanga.