On the Topological Structure and Properties of Multidimensional (C, R) Space
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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. -
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. -
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*
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. -
Arxiv:Math/0405137V1 [Math.RT] 7 May 2004 Oino Disbecniuu Ersnain Fsc Group)
Draft: May 4, 2004 LOCALLY ANALYTIC VECTORS IN REPRESENTATIONS OF LOCALLY p-ADIC ANALYTIC GROUPS Matthew Emerton Northwestern University Contents 0. Terminology, notation and conventions 7 1. Non-archimedean functional analysis 10 2. Non-archimedean function theory 27 3. Continuous, analytic, and locally analytic vectors 43 4. Smooth, locally finite, and locally algebraic vectors 71 5. Rings of distributions 81 6. Admissible locally analytic representations 100 7. Representationsofcertainproductgroups 124 References 135 Recent years have seen the emergence of a new branch of representation theory: the theory of representations of locally p-adic analytic groups on locally convex p-adic topological vector spaces (or “locally analytic representation theory”, for short). Examples of such representations are provided by finite dimensional alge- braic representations of p-adic reductive groups, and also by smooth representations of such groups (on p-adic vector spaces). One might call these the “classical” ex- amples of such representations. One of the main interests of the theory (from the point of view of number theory) is that it provides a setting in which one can study p-adic completions of the classical representations [6], or construct “p-adic interpolations” of them (for example, by defining locally analytic analogues of the principal series, as in [20], or by constructing representations via the cohomology of arithmetic quotients of symmetric spaces, as in [9]). Locally analytic representation theory also plays an important role in the analysis of p-adic symmetric spaces; indeed, this analysis provided the original motivation for its development. The first “non-classical” examples in the theory were found by Morita, in his analysis of the p-adic upper half-plane (the p-adic symmetric arXiv:math/0405137v1 [math.RT] 7 May 2004 space attached to GL2(Qp)) [15], and further examples were found by Schneider and Teitelbaum in their analytic investigations of the p-adic symmetric spaces of GLn(Qp) (for arbitrary n) [19]. -
Spectral Radii of Bounded Operators on Topological Vector Spaces
SPECTRAL RADII OF BOUNDED OPERATORS ON TOPOLOGICAL VECTOR SPACES VLADIMIR G. TROITSKY Abstract. In this paper we develop a version of spectral theory for bounded linear operators on topological vector spaces. We show that the Gelfand formula for spectral radius and Neumann series can still be naturally interpreted for operators on topological vector spaces. Of course, the resulting theory has many similarities to the conventional spectral theory of bounded operators on Banach spaces, though there are several im- portant differences. The main difference is that an operator on a topological vector space has several spectra and several spectral radii, which fit a well-organized pattern. 0. Introduction The spectral radius of a bounded linear operator T on a Banach space is defined by the Gelfand formula r(T ) = lim n T n . It is well known that r(T ) equals the n k k actual radius of the spectrum σ(T ) p= sup λ : λ σ(T ) . Further, it is known −1 {| | ∈ } ∞ T i that the resolvent R =(λI T ) is given by the Neumann series +1 whenever λ − i=0 λi λ > r(T ). It is natural to ask if similar results are valid in a moreP general setting, | | e.g., for a bounded linear operator on an arbitrary topological vector space. The author arrived to these questions when generalizing some results on Invariant Subspace Problem from Banach lattices to ordered topological vector spaces. One major difficulty is that it is not clear which class of operators should be considered, because there are several non-equivalent ways of defining bounded operators on topological vector spaces. -
Sufficient Generalities About Topological Vector Spaces
(November 28, 2016) Topological vector spaces Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/~garrett/m/fun/notes 2016-17/tvss.pdf] 1. Banach spaces Ck[a; b] 2. Non-Banach limit C1[a; b] of Banach spaces Ck[a; b] 3. Sufficient notion of topological vector space 4. Unique vectorspace topology on Cn 5. Non-Fr´echet colimit C1 of Cn, quasi-completeness 6. Seminorms and locally convex topologies 7. Quasi-completeness theorem 1. Banach spaces Ck[a; b] We give the vector space Ck[a; b] of k-times continuously differentiable functions on an interval [a; b] a metric which makes it complete. Mere pointwise limits of continuous functions easily fail to be continuous. First recall the standard [1.1] Claim: The set Co(K) of complex-valued continuous functions on a compact set K is complete with o the metric jf − gjCo , with the C -norm jfjCo = supx2K jf(x)j. Proof: This is a typical three-epsilon argument. To show that a Cauchy sequence ffig of continuous functions has a pointwise limit which is a continuous function, first argue that fi has a pointwise limit at every x 2 K. Given " > 0, choose N large enough such that jfi − fjj < " for all i; j ≥ N. Then jfi(x) − fj(x)j < " for any x in K. Thus, the sequence of values fi(x) is a Cauchy sequence of complex numbers, so has a limit 0 0 f(x). Further, given " > 0 choose j ≥ N sufficiently large such that jfj(x) − f(x)j < " . -
Note on the Solvability of Equations Involving Unbounded Linear and Quasibounded Nonlinear Operators*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 56, 495-501 (1976) Note on the Solvability of Equations Involving Unbounded Linear and Quasibounded Nonlinear Operators* W. V. PETRYSHYN Deportment of Mathematics, Rutgers University, New Brunswick, New Jersey 08903 Submitted by Ky Fan INTRODUCTION Let X and Y be Banach spaces, X* and Y* their respective duals and (u, x) the value of u in X* at x in X. For a bounded linear operator T: X---f Y (i.e., for T EL(X, Y)) let N(T) C X and R(T) C Y denote the null space and the range of T respectively and let T *: Y* --f X* denote the adjoint of T. Foranyset vinXandWinX*weset I”-={uEX*~(U,X) =OV~EV) and WL = {x E X 1(u, x) = OVu E W). In [6] Kachurovskii obtained (without proof) a partial generalization of the closed range theorem of Banach for mildly nonlinear equations which can be stated as follows: THEOREM K. Let T EL(X, Y) have a closed range R(T) and a Jinite dimensional null space N(T). Let S: X --f Y be a nonlinear compact mapping such that R(S) C N( T*)l and S is asymptotically zer0.l Then the equation TX + S(x) = y is solvable for a given y in Y if and only if y E N( T*)l. The purpose of this note is to extend Theorem K to equations of the form TX + S(x) = y (x E WY, y E Y), (1) where T: D(T) X--f Y is a closed linear operator with domain D(T) not necessarily dense in X and such that R(T) is closed and N(T) (CD(T)) has a closed complementary subspace in X and S: X-t Y is a nonlinear quasi- bounded mapping but not necessarily compact or asymptotically zero. -
Lp Spaces in Isabelle
Lp spaces in Isabelle Sebastien Gouezel Abstract Lp is the space of functions whose p-th power is integrable. It is one of the most fundamental Banach spaces that is used in analysis and probability. We develop a framework for function spaces, and then im- plement the Lp spaces in this framework using the existing integration theory in Isabelle/HOL. Our development contains most fundamental properties of Lp spaces, notably the Hölder and Minkowski inequalities, completeness of Lp, duality, stability under almost sure convergence, multiplication of functions in Lp and Lq, stability under conditional expectation. Contents 1 Functions as a real vector space 2 2 Quasinorms on function spaces 4 2.1 Definition of quasinorms ..................... 5 2.2 The space and the zero space of a quasinorm ......... 7 2.3 An example: the ambient norm in a normed vector space .. 9 2.4 An example: the space of bounded continuous functions from a topological space to a normed real vector space ....... 10 2.5 Continuous inclusions between functional spaces ....... 10 2.6 The intersection and the sum of two functional spaces .... 12 2.7 Topology ............................. 14 3 Conjugate exponents 16 4 Convexity inequalities and integration 18 5 Lp spaces 19 5.1 L1 ................................. 20 5.2 Lp for 0 < p < 1 ......................... 21 5.3 Specialization to L1 ....................... 22 5.4 L0 ................................. 23 5.5 Basic results on Lp for general p ................ 24 5.6 Lp versions of the main theorems in integration theory .... 25 1 5.7 Completeness of Lp ........................ 26 5.8 Multiplication of functions, duality ............... 26 5.9 Conditional expectations and Lp ............... -
On Almost S-Topological Vector Spaces
Journal of Advanced Studies in Topology 9:2 (2018), 139–146 On almost s-topological vector spaces a, a a a Madhu Ram ∗, Shallu Sharma , Sahil Billawria , Amjad Hussain aDepartment of Mathematics, University of Jammu, JK-180006, India. Abstract In this paper, we introduce the notion of almost s-topological vector spaces and present some examples and counterexamples of almost s-topological vector spaces. Some general properties of almost s-topological vector spaces are investigated. Keywords: Semi-open sets, regular open sets, almost s-topological vector spaces. 2010 MSC: 57N17, 57N99. 1. Introduction The concept of topological vector spaces was first given by Kolmogroff [4] in 1934. Since then, different researchers have been explored many interesting and useful properties of topological vector spaces. Due to nice properties, topological vector spaces earn a great importance and remain a fundamental notion in fixed point theory, operator theory, variational inequalities, etc. Nowadays the researchers not only make use of topological vector spaces in other fields for developing new notions but also stretch and extend the concept of topological vector spaces with every possible way, making this field of study a more convenient and understandable. Recently, M. Khan et al. [2] introduced the s-topological vector spaces; a generalization of topological vector spaces. In 2016, Khan and Iqbal [3] introduced the irresolute topological vector spaces which are a particular brand of s-topological vector spaces but irresolute topological vector spaces are independent of topological vector spaces. Ibrahim [1] initiated the study of α topological vector spaces. − S. Sharma et. al. [7] put forth the concept of almost pretopological vector spaces.