Sequential Convergence in Topological Vector Spaces
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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. -
Boundedness in Linear Topological Spaces
BOUNDEDNESS IN LINEAR TOPOLOGICAL SPACES BY S. SIMONS Introduction. Throughout this paper we use the symbol X for a (real or complex) linear space, and the symbol F to represent the basic field in question. We write R+ for the set of positive (i.e., ^ 0) real numbers. We use the term linear topological space with its usual meaning (not necessarily Tx), but we exclude the case where the space has the indiscrete topology (see [1, 3.3, pp. 123-127]). A linear topological space is said to be a locally bounded space if there is a bounded neighbourhood of 0—which comes to the same thing as saying that there is a neighbourhood U of 0 such that the sets {(1/n) U} (n = 1,2,—) form a base at 0. In §1 we give a necessary and sufficient condition, in terms of invariant pseudo- metrics, for a linear topological space to be locally bounded. In §2 we discuss the relationship of our results with other results known on the subject. In §3 we introduce two ways of classifying the locally bounded spaces into types in such a way that each type contains exactly one of the F spaces (0 < p ^ 1), and show that these two methods of classification turn out to be identical. Also in §3 we prove a metrization theorem for locally bounded spaces, which is related to the normal metrization theorem for uniform spaces, but which uses a different induction procedure. In §4 we introduce a large class of linear topological spaces which includes the locally convex spaces and the locally bounded spaces, and for which one of the more important results on boundedness in locally convex spaces is valid. -
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. -
I-Sequential Topological Spaces∗
Applied Mathematics E-Notes, 14(2014), 236-241 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ -Sequential Topological Spaces I Sudip Kumar Pal y Received 10 June 2014 Abstract In this paper a new notion of topological spaces namely, I-sequential topo- logical spaces is introduced and investigated. This new space is a strictly weaker notion than the first countable space. Also I-sequential topological space is a quotient of a metric space. 1 Introduction The idea of convergence of real sequence have been extended to statistical convergence by [2, 14, 15] as follows: If N denotes the set of natural numbers and K N then Kn denotes the set k K : k n and Kn stands for the cardinality of the set Kn. The natural densityf of the2 subset≤ Kg is definedj j by Kn d(K) = lim j j, n n !1 provided the limit exists. A sequence xn n N of points in a metric space (X, ) is said to be statistically f g 2 convergent to l if for arbitrary " > 0, the set K(") = k N : (xk, l) " has natural density zero. A lot of investigation has been donef on2 this convergence≥ andg its topological consequences after initial works by [5, 13]. It is easy to check that the family Id = A N : d(A) = 0 forms a non-trivial f g admissible ideal of N (recall that I 2N is called an ideal if (i) A, B I implies A B I and (ii) A I,B A implies B I. -
On Compact-Covering and Sequence-Covering Images of Metric Spaces
MATEMATIQKI VESNIK originalni nauqni rad 64, 2 (2012), 97–107 research paper June 2012 ON COMPACT-COVERING AND SEQUENCE-COVERING IMAGES OF METRIC SPACES Jing Zhang Abstract. In this paper we study the characterizations of compact-covering and 1-sequence- covering (resp. 2-sequence-covering) images of metric spaces and give a positive answer to the following question: How to characterize first countable spaces whose each compact subset is metrizable by certain images of metric spaces? 1. Introduction To find internal characterizations of certain images of metric spaces is one of the central problems in General Topology. In 1973, E. Michael and K. Nagami [16] obtained a characterization of compact-covering and open images of metric spaces. It is well known that the compact-covering and open images of metric spaces are the first countable spaces whose each compact subset is metrizable. However, its inverse does not hold [16]. For the first countable spaces whose each compact subset is metrizable, how to characterize them by certain images of metric spaces? The sequence-covering maps play an important role on mapping theory about metric spaces [6, 9]. In this paper, we give the characterization of a compact- covering and 1-sequence-covering (resp. 2-sequence-covering) image of a metric space, and positively answer the question posed by S. Lin in [11, Question 2.6.5]. All spaces considered here are T2 and all maps are continuous and onto. The letter N is the set of all positive natural numbers. Readers may refer to [10] for unstated definition and terminology. 2. -
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 . -
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. -
Arxiv:1702.07867V1 [Math.FA]
TOPOLOGICAL PROPERTIES OF STRICT (LF )-SPACES AND STRONG DUALS OF MONTEL STRICT (LF )-SPACES SAAK GABRIYELYAN Abstract. Following [2], a Tychonoff space X is Ascoli if every compact subset of Ck(X) is equicontinuous. By the classical Ascoli theorem every k- space is Ascoli. We show that a strict (LF )-space E is Ascoli iff E is a Fr´echet ′ space or E = ϕ. We prove that the strong dual Eβ of a Montel strict (LF )- space E is an Ascoli space iff one of the following assertions holds: (i) E is a ′ Fr´echet–Montel space, so Eβ is a sequential non-Fr´echet–Urysohn space, or (ii) ′ ω D E = ϕ, so Eβ = R . Consequently, the space (Ω) of test functions and the space of distributions D′(Ω) are not Ascoli that strengthens results of Shirai [20] and Dudley [5], respectively. 1. Introduction. The class of strict (LF )-spaces was intensively studied in the classic paper of Dieudonn´eand Schwartz [3]. It turns out that many of strict (LF )-spaces, in particular a lot of linear spaces considered in Schwartz’s theory of distributions [18], are not metrizable. Even the simplest ℵ0-dimensional strict (LF )-space ϕ, n the inductive limit of the sequence {R }n∈ω, is not metrizable. Nyikos [16] showed that ϕ is a sequential non-Fr´echet–Urysohn space (all relevant definitions are given in the next section). On the other hand, Shirai [20] proved the space D(Ω) of test functions over an open subset Ω of Rn, which is one of the most famous example of strict (LF )-spaces, is not sequential. -
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. -
On Nonbornological Barrelled Spaces Annales De L’Institut Fourier, Tome 22, No 2 (1972), P
ANNALES DE L’INSTITUT FOURIER MANUEL VALDIVIA On nonbornological barrelled spaces Annales de l’institut Fourier, tome 22, no 2 (1972), p. 27-30 <http://www.numdam.org/item?id=AIF_1972__22_2_27_0> © Annales de l’institut Fourier, 1972, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Fourier, Grenoble 22, 2 (1972), 27-30. ON NONBORNOLOGICAL BARRELLED SPACES 0 by Manuel VALDIVIA L. Nachbin [5] and T. Shirota [6], give an answer to a problem proposed by N. Bourbaki [1] and J. Dieudonne [2], giving an example of a barrelled space, which is not bornological. Posteriorly some examples of nonbornological barrelled spaces have been given, e.g. Y. Komura, [4], has constructed a Montel space which is not bornological. In this paper we prove that if E is the topological product of an uncountable family of barrelled spaces, of nonzero dimension, there exists an infinite number of barrelled subspaces of E, which are not bornological.We obtain also an analogous result replacing « barrelled » by « quasi-barrelled ». We use here nonzero vector spaces on the field K of real or complex number. The topologies on these spaces are sepa- rated. -
Fact Sheet Functional Analysis
Fact Sheet Functional Analysis Literature: Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen. Teubner, 1986. Knabner, P., Angermann, L.: Numerik partieller Differentialgleichungen. Springer, 2000. Triebel, H.: H¨ohere Analysis. Harri Deutsch, 1980. Dobrowolski, M.: Angewandte Funktionalanalysis, Springer, 2010. 1. Banach- and Hilbert spaces Let V be a real vector space. Normed space: A norm is a mapping k · k : V ! [0; 1), such that: kuk = 0 , u = 0; (definiteness) kαuk = jαj · kuk; α 2 R; u 2 V; (positive scalability) ku + vk ≤ kuk + kvk; u; v 2 V: (triangle inequality) The pairing (V; k · k) is called a normed space. Seminorm: In contrast to a norm there may be elements u 6= 0 such that kuk = 0. It still holds kuk = 0 if u = 0. Comparison of two norms: Two norms k · k1, k · k2 are called equivalent if there is a constant C such that: −1 C kuk1 ≤ kuk2 ≤ Ckuk1; u 2 V: If only one of these inequalities can be fulfilled, e.g. kuk2 ≤ Ckuk1; u 2 V; the norm k · k1 is called stronger than the norm k · k2. k · k2 is called weaker than k · k1. Topology: In every normed space a canonical topology can be defined. A subset U ⊂ V is called open if for every u 2 U there exists a " > 0 such that B"(u) = fv 2 V : ku − vk < "g ⊂ U: Convergence: A sequence vn converges to v w.r.t. the norm k · k if lim kvn − vk = 0: n!1 1 A sequence vn ⊂ V is called Cauchy sequence, if supfkvn − vmk : n; m ≥ kg ! 0 for k ! 1.