COMPLEMENTED SUBALGEBRAS OF THE BAIRE-1 FUNCTIONS DEFINED ON THE INTERVAL [0,1]
|
|
- Drusilla Pierce
- 6 years ago
- Views:
Transcription
1 COMPLEMENTED SUBALGEBRAS OF THE BAIRE- FUNCTIONS DEFINED ON THE INTERVAL [0,] H. R. SHATERY Received 27 April 2004 and in revised form 26 September 2004 We prove that if the Banach algebra of bounded real Baire- functions (resp., small Baire class ξ, defined on [0,], is the direct sum of two subalgebras, then one of its components contains a copy of it as a complemented subalgebra. An important problem in topology is to determine the effects on the function space of imposing some natural topological condition on the space X [5]. In this way, it is practical to characterize the complemented subalgebras of Banach algebras. In our investigation, we relate the second category property of [0,] with certain properties of the complemented subalgebras of bounded real Baire- functions, β ([0,] (bounded functions of finite Baire index, Ϝ ([0,]. We begin by recalling some definitions. Let A be a Banach algebra (resp., Banach space. Two subalgebras (resp., subspaces M and N of A are complementary if A = M N. A projection on A is a continuous linear operator P : A A satisfying P 2 = P. If M and N are complementary subalgebras (resp., subspaces of A, then there exists a projection P on A such that the range of P is M (N. The norm (sup-norm of the projection P is always equal to or greater than. Norm projections play a crucial role in the study of complemented subalgebras (resp., subspaces of Banach algebras (resp., Banach spaces (see, e.g., [4]. If A is a finite-dimensional Banach space, then every nontrivial subspace of A is closed and complemented in A and does not contain a copy of A. This of course is more complicated for infinite-dimensional Banach spaces. Pelczynski has proved that every infinite-dimensional closed linear subspace of l contains a complemented subspace of l that is isomorphic to l [4, Theorem 6, page 74]. Also it has been proved that C(X, the ring of continuous functions on the compact topological space X is the direct sum of two proper subrings if and only if X is disconnected [5, Problem.B, page 20]. In this case, for every decomposition of C(X, there is an open compact partition {A,B} of X such that C(X = C(A C(B. We want to establish a result similar to that of Pelczynski for the Banach algebra of bounded real Baire- functions defined on [0,]. Copyright 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005:3 ( DOI: 0.55/IJMMS
2 446 Complemented subalgebras of the Baire- functions Throughout this paper, X is a compact subset of real numbers. The class of open (resp., closed subsets of X is denoted by (resp.,. We define δ (resp., σ astheclassofall of countable intersections (resp., unions of elements of (resp.,. We denote the set ( δ σ by. Let X be a topological space. We define the real Baire functions of class as follows: β (X = { f : X R : ( f n n= C(X such that lim f n (x = f (x, for each x X }. ( We denote the set of bounded functions in β (X byβ (X. The Baire- class, β (X has an algebraic and isometric representation as the space C(ω of all continuous functions on a totally disconnected compact space ω. This representation was used to show that if the compact space S has an uncountable compact metrizable subset, then β (Sis not linearly isomorphic to any complemented subspace of the Banach space C(K for σ-stonian space K [3]. In [], Bade studied the linear complementation problem for the Baire classes. He proved that β α ([0,] is not complemented as a closed subspace of β α+ ([0,] for each ordinal α<ω. In our investigation, we characterize the complemented topological subalgebras of the Baire- classes on [0,]. Theorem. If the Banach algebra of bounded real Baire- functions, defined on [0,], is the direct sum of two subalgebras, then one of its components contains a copy of it as a complemented subalgebra; that is, if A and B are two subalgebras of β ([0,] such that then there exist two subalgebras, C and D,ofA (or B such that β ( [0,] = A B, (2 A = C D, C = β ( [0,]. (3 Moreover, each complemented subalgebra of β ([0,] can be obtained by a norm,positive and multiplicative projection. Proof. First we note that the idempotents of the ring β ([0,] are χ H s for H []. It is obvious that H if and only if [0,] H. Suppose that the ring β ([0,] is the direct sum of two subrings A and B, β ( [0,] = A B. (4 The constant function ˆ belongstoβ ([0,], therefore there exist e A and e 2 B such that ˆ = e + e 2. (5 Thus e and e 2 are two disjoint idempotents; that is, e e 2 = 0, because A B ={0},and e e 2 = e e 2 = e 2 e2 2 (A B ={0}. (6
3 H. R. Shatery 447 Suppose that e = χ H and e 2 = χ H2 for suitable H and H 2 in. By(5, {H,H 2 } is a partition of [0,] by sets. Therefore, we conclude that A = β([0,] χ H = β (χ H and similarly B = β(χ H2. Now, suppose that {H,H 2 } is a partition of [0,] by sets. Let f and g be in β(h andβ(h 2, respectively. We define h as follows: f (x ifx H, h(x = (7 g(x ifx H 2. Suppose F is a closed subset of R.Then h (F = f (F g (F. (8 Hence f (F andg (F are δ sets in H and H 2, respectively, and so they are both δ in [0,]. Consequently h (F is δ in [0,] and h β (X []. It is obvious that if h β (X, then h H β (H andh H2 β (H 2. We now define ϕ from β ([0,] onto β (H β (H 2 as ϕ( f = ( f H, f H2. (9 Then ϕ is a surjective algebra isomorphism. Thus there exists a one-to-one correspondence between algebra decompositions of β ([0,] and the partitions of [0,]. The interval [0,] is a second-category topological space, and H and H 2 are countable unions of closed sets, therefore one of them has nonempty interior. Suppose the interior of H is not empty. Then there exists a closed interval [a,b] H.Clearly,wehave β ( [0,] = β ( [a,b]. (0 On the other hand, there exists an set, H 3 in [0,] disjoint from [a,b]suchthat Therefore, H = [a,b] H 3. ( β ( H = β ( [a,b] β ( H3. (2 Thus β ([0,] is complemented in β (H. Now we prove the second assertion. Let C be a complemented subalgebra of β ([0,]. Therefore, there exists an set, H in [0,] such that C = β (H. We define ( P : β( [0,] β [0,], P( f = f H. (3 Let H c = [0,] H and f β([0,]. We have f = f H + f H c.thus, f =max ( f H, f H c f H. (4 Therefore, P. It follows that P = since the norm of a projection is always equal to or greater than. If f β([0,] and f 0, then f H 0, and therefore, P is positive. Also, it is obvious that P is multiplicative. The proof is now complete.
4 448 Complemented subalgebras of the Baire- functions The finite and small Baire classes have been studied by many people (e.g., [2, 7]. We begin by recalling the definition of the index β. Suppose that H is an set in [0,], and f is a real-valued function whose domain is H.Foranyɛ > 0, let H 0 ( f,ɛ = H.IfH α ( f,ɛ is defined for some countable ordinal α,leth α+ ( f,ɛ be the set of all those x H α ( f,ɛ such that for every open U containing x, there are two points x and x 2 in U H α ( f,ɛ with f (x f (x 2 ɛ. For a countable limit ordinal α,welet H α ( f,ɛ = α <αh α ( f,ɛ. (5 The index β H ( f,ɛ is taken to be the least α with H α ( f,ɛ = if such α exists, and ω otherwise. The oscillation index of f is β H ( f = sup { β H ( f,ɛ:ɛ > 0 }. (6 It is known that f : H R is Baire- if and only if β H ( f <ω [7]. We define the set of bounded functions of finite Baire index (resp., small Baire class ξ for each countable ordinal ξas Ϝ (H = { f β(h: β H ( f < } ( ξ, (H = { f β(h: β H ( f ω ξ}. (7 The set of bounded functions of finite Baire index, Ϝ (resp., small Baire class ξ, ξ (H, is a Banach algebra (with sup-norm. It is obvious that if f : H R is a Baire- function, H 2 H [0,] (H,H 2 andg = f H2,thenβ H2 (g β H ( f. Therefore, if f Ϝ (resp., f ξ (H, then g Ϝ (resp., g ξ (H 2 (but the converse is not true. So we have the following. Remark 2. The previous theorem is also valid for the set of bounded functions of finite Baire index (resp., small Baire class ξ. It may happen that for two sets, H and H 2 ( [0,] such that H 2 = [0,] H, and f Ϝ (H and f 2 Ϝ (H 2, f = f + f 2 does not belong to Ϝ ([0,]. Suppose that H is the Cantor set C in the interval [0,] and H 2 = [0,] C.Let f i = iχ Hi (i =,2, and f = f + f 2. It is obvious that f / Ϝ ([0,]. Thus, ( [0,] Ϝ( H Ϝ( H2. (8 Ϝ But if M and N are two complementary subalgebras of ξ ([0,], then there exist suitable sets, H and H 2,suchthat M = Ϝ ( H, N = Ϝ( H2. (9 By the above argument and using the epimorphism Θ : Ϝ ( [0,] Ϝ (C, Θ( f = f C, (20
5 H. R. Shatery 449 we see that there exists a noncomplemented subalgebra such that Ϝ ( [0,] = Ϝ (C, (2 and is not of the form Ϝ (Hforany set H. (The algebra homomorphism Θ is onto by Tietze extension theorem [8, Theorem 3.6]. It has been proved that for real compact spaces X and Y, a linear isometry between β (X andβ (Y induces an algebra (a ring isometry [6]. Is this true for linear complemented subspaces of β ([0,]? If the answer to the above question is positive, then it should be easy to prove an analogous theorem for linear complemented subspaces of β ([0,]. Acknowledgment The author is grateful to the referees for their helpful comments. References [] W. G. Bade, Complementation problems for the Baire classes,pacific J. Math. 45 (973,. [2] F. Chaatit, V. Mascioni, and H. Rosenthal, On functions of finite Baire index, J.Funct. Anal.42 (996, no. 2, [3] F. K. DashiellJr.,Isomorphism problems for the Baire classes,pacific J. Math. 52 (974, [4] J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 984. [5] L. GillmanandM. Jerison, Rings of Continuous Functions, The University Series in Higher Mathematics, D. Van Nostrand, New Jersey, 960. [6] J. E. Jayne, The space of class α Baire functions, Bull. Amer. Math. Soc. 80 (974, [7] A.S.KechrisandA.Louveau,A classification of Baire class functions,trans.amer.math.soc. 38 (990, no., [8] D. H. Leung and W.-K. Tang, Functions of Baire class one, Fund. Math. 79 (2003, no. 3, H. R. Shatery: Department of Mathematics, University of Isfahan, Isfahan, Iran address: shatery@math.ui.ac.ir
Problem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationMH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then
MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever
More informationMath 172 HW 1 Solutions
Math 172 HW 1 Solutions Joey Zou April 15, 2017 Problem 1: Prove that the Cantor set C constructed in the text is totally disconnected and perfect. In other words, given two distinct points x, y C, there
More informationSOME BANACH SPACE GEOMETRY
SOME BANACH SPACE GEOMETRY SVANTE JANSON 1. Introduction I have collected some standard facts about Banach spaces from various sources, see the references below for further results. Proofs are only given
More informationVARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES
Bull. Austral. Math. Soc. 78 (2008), 487 495 doi:10.1017/s0004972708000877 VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES CAROLYN E. MCPHAIL and SIDNEY A. MORRIS (Received 3 March 2008) Abstract
More informationFragmentability and σ-fragmentability
F U N D A M E N T A MATHEMATICAE 143 (1993) Fragmentability and σ-fragmentability by J. E. J a y n e (London), I. N a m i o k a (Seattle) and C. A. R o g e r s (London) Abstract. Recent work has studied
More informationP.S. Gevorgyan and S.D. Iliadis. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 70, 2 (208), 0 9 June 208 research paper originalni nauqni rad GROUPS OF GENERALIZED ISOTOPIES AND GENERALIZED G-SPACES P.S. Gevorgyan and S.D. Iliadis Abstract. The
More informationREPRESENTABLE BANACH SPACES AND UNIFORMLY GÂTEAUX-SMOOTH NORMS. Julien Frontisi
Serdica Math. J. 22 (1996), 33-38 REPRESENTABLE BANACH SPACES AND UNIFORMLY GÂTEAUX-SMOOTH NORMS Julien Frontisi Communicated by G. Godefroy Abstract. It is proved that a representable non-separable Banach
More informationvan Rooij, Schikhof: A Second Course on Real Functions
vanrooijschikhof.tex April 25, 2018 van Rooij, Schikhof: A Second Course on Real Functions Notes from [vrs]. Introduction A monotone function is Riemann integrable. A continuous function is Riemann integrable.
More informationReal Analysis, 2nd Edition, G.B.Folland Elements of Functional Analysis
Real Analysis, 2nd Edition, G.B.Folland Chapter 5 Elements of Functional Analysis Yung-Hsiang Huang 5.1 Normed Vector Spaces 1. Note for any x, y X and a, b K, x+y x + y and by ax b y x + b a x. 2. It
More information2.2 Annihilators, complemented subspaces
34CHAPTER 2. WEAK TOPOLOGIES, REFLEXIVITY, ADJOINT OPERATORS 2.2 Annihilators, complemented subspaces Definition 2.2.1. [Annihilators, Pre-Annihilators] Assume X is a Banach space. Let M X and N X. We
More information3. Prove or disprove: If a space X is second countable, then every open covering of X contains a countable subcollection covering X.
Department of Mathematics and Statistics University of South Florida TOPOLOGY QUALIFYING EXAM January 24, 2015 Examiners: Dr. M. Elhamdadi, Dr. M. Saito Instructions: For Ph.D. level, complete at least
More informationCHAPTER V DUAL SPACES
CHAPTER V DUAL SPACES DEFINITION Let (X, T ) be a (real) locally convex topological vector space. By the dual space X, or (X, T ), of X we mean the set of all continuous linear functionals on X. By the
More informationSOLUTIONS TO THE FINAL EXAM
SOLUTIONS TO THE FINAL EXAM Short questions 1 point each) Give a brief definition for each of the following six concepts: 1) normal for topological spaces) 2) path connected 3) homeomorphism 4) covering
More informationDAVIS WIELANDT SHELLS OF NORMAL OPERATORS
DAVIS WIELANDT SHELLS OF NORMAL OPERATORS CHI-KWONG LI AND YIU-TUNG POON Dedicated to Professor Hans Schneider for his 80th birthday. Abstract. For a finite-dimensional operator A with spectrum σ(a), the
More informationANALYSIS WORKSHEET II: METRIC SPACES
ANALYSIS WORKSHEET II: METRIC SPACES Definition 1. A metric space (X, d) is a space X of objects (called points), together with a distance function or metric d : X X [0, ), which associates to each pair
More information1 Measure and Category on the Line
oxtoby.tex September 28, 2011 Oxtoby: Measure and Category Notes from [O]. 1 Measure and Category on the Line Countable sets, sets of first category, nullsets, the theorems of Cantor, Baire and Borel Theorem
More informationLocally uniformly rotund renormings of the spaces of continuous functions on Fedorchuk compacts
Locally uniformly rotund renormings of the spaces of continuous functions on Fedorchuk compacts S.TROYANSKI, Institute of Mathematics, Bulgarian Academy of Science joint work with S. P. GUL KO, Faculty
More information7 Complete metric spaces and function spaces
7 Complete metric spaces and function spaces 7.1 Completeness Let (X, d) be a metric space. Definition 7.1. A sequence (x n ) n N in X is a Cauchy sequence if for any ɛ > 0, there is N N such that n, m
More informationOn α-embedded subsets of products
European Journal of Mathematics 2015) 1:160 169 DOI 10.1007/s40879-014-0018-0 RESEARCH ARTICLE On α-embedded subsets of products Olena Karlova Volodymyr Mykhaylyuk Received: 22 May 2014 / Accepted: 19
More informationarxiv: v1 [math.gn] 27 Oct 2018
EXTENSION OF BOUNDED BAIRE-ONE FUNCTIONS VS EXTENSION OF UNBOUNDED BAIRE-ONE FUNCTIONS OLENA KARLOVA 1 AND VOLODYMYR MYKHAYLYUK 1,2 Abstract. We compare possibilities of extension of bounded and unbounded
More informationFUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES. Yılmaz Yılmaz
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 33, 2009, 335 353 FUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES Yılmaz Yılmaz Abstract. Our main interest in this
More informationA normally supercompact Parovičenko space
A normally supercompact Parovičenko space A. Kucharski University of Silesia, Katowice, Poland Hejnice, 26.01-04.02 2013 normally supercompact Parovičenko space Hejnice, 26.01-04.02 2013 1 / 12 These results
More informationExtensions of Lipschitz functions and Grothendieck s bounded approximation property
North-Western European Journal of Mathematics Extensions of Lipschitz functions and Grothendieck s bounded approximation property Gilles Godefroy 1 Received: January 29, 2015/Accepted: March 6, 2015/Online:
More informationReal Analysis Notes. Thomas Goller
Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................
More informationISOMORPHISM PROBLEMS FOR THE BAIRE FUNCTION SPACES OF TOPOLOGICAL SPACES. Mitrofan M. Choban
Serdica Math. J. 24 (1998), 5-20 ISOMORPHISM PROBLEMS FOR THE BAIRE FUNCTION SPACES OF TOPOLOGICAL SPACES Mitrofan M. Choban Communicated by J. Jayne Dedicated to the memory of Professor D. Doitchinov
More informationCOUNTEREXAMPLES IN ROTUND AND LOCALLY UNIFORMLY ROTUND NORM
Journal of Mathematical Analysis ISSN: 2217-3412, URL: http://www.ilirias.com Volume 5 Issue 1(2014), Pages 11-15. COUNTEREXAMPLES IN ROTUND AND LOCALLY UNIFORMLY ROTUND NORM F. HEYDARI, D. BEHMARDI 1
More informationThe small ball property in Banach spaces (quantitative results)
The small ball property in Banach spaces (quantitative results) Ehrhard Behrends Abstract A metric space (M, d) is said to have the small ball property (sbp) if for every ε 0 > 0 there exists a sequence
More informationIntroduction to Functional Analysis
Introduction to Functional Analysis Carnegie Mellon University, 21-640, Spring 2014 Acknowledgements These notes are based on the lecture course given by Irene Fonseca but may differ from the exact lecture
More information1.A Topological spaces The initial topology is called topology generated by (f i ) i I.
kechris.tex December 12, 2012 Classical descriptive set theory Notes from [Ke]. 1 1 Polish spaces 1.1 Topological and metric spaces 1.A Topological spaces The initial topology is called topology generated
More informationAnalysis Comprehensive Exam Questions Fall F(x) = 1 x. f(t)dt. t 1 2. tf 2 (t)dt. and g(t, x) = 2 t. 2 t
Analysis Comprehensive Exam Questions Fall 2. Let f L 2 (, ) be given. (a) Prove that ( x 2 f(t) dt) 2 x x t f(t) 2 dt. (b) Given part (a), prove that F L 2 (, ) 2 f L 2 (, ), where F(x) = x (a) Using
More informationMath General Topology Fall 2012 Homework 11 Solutions
Math 535 - General Topology Fall 2012 Homework 11 Solutions Problem 1. Let X be a topological space. a. Show that the following properties of a subset A X are equivalent. 1. The closure of A in X has empty
More informationTopological Properties of Operations on Spaces of Continuous Functions and Integrable Functions
Topological Properties of Operations on Spaces of Continuous Functions and Integrable Functions Holly Renaud University of Memphis hrenaud@memphis.edu May 3, 2018 Holly Renaud (UofM) Topological Properties
More informationBoundedly complete weak-cauchy basic sequences in Banach spaces with the PCP
Journal of Functional Analysis 253 (2007) 772 781 www.elsevier.com/locate/jfa Note Boundedly complete weak-cauchy basic sequences in Banach spaces with the PCP Haskell Rosenthal Department of Mathematics,
More informationA COMMENT ON FREE GROUP FACTORS
A COMMENT ON FREE GROUP FACTORS NARUTAKA OZAWA Abstract. Let M be a finite von Neumann algebra acting on the standard Hilbert space L 2 (M). We look at the space of those bounded operators on L 2 (M) that
More informationOn z -ideals in C(X) F. A z a r p a n a h, O. A. S. K a r a m z a d e h and A. R e z a i A l i a b a d (Ahvaz)
F U N D A M E N T A MATHEMATICAE 160 (1999) On z -ideals in C(X) by F. A z a r p a n a h, O. A. S. K a r a m z a d e h and A. R e z a i A l i a b a d (Ahvaz) Abstract. An ideal I in a commutative ring
More informationCOUNTABLY S-CLOSED SPACES
COUNTABLY S-CLOSED SPACES Karin DLASKA, Nurettin ERGUN and Maximilian GANSTER Abstract In this paper we introduce the class of countably S-closed spaces which lies between the familiar classes of S-closed
More informationAliprantis, Border: Infinite-dimensional Analysis A Hitchhiker s Guide
aliprantis.tex May 10, 2011 Aliprantis, Border: Infinite-dimensional Analysis A Hitchhiker s Guide Notes from [AB2]. 1 Odds and Ends 2 Topology 2.1 Topological spaces Example. (2.2) A semimetric = triangle
More informationRoots and Root Spaces of Compact Banach Lie Algebras
Irish Math. Soc. Bulletin 49 (2002), 15 22 15 Roots and Root Spaces of Compact Banach Lie Algebras A. J. CALDERÓN MARTÍN AND M. FORERO PIULESTÁN Abstract. We study the main properties of roots and root
More informationContinuous functions with compact support
@ Applied General Topology c Universidad Politécnica de Valencia Volume 5, No. 1, 2004 pp. 103 113 Continuous functions with compact support S. K. Acharyya, K. C. Chattopadhyaya and Partha Pratim Ghosh
More informationRecursion Theoretic Methods in Descriptive Set Theory and Infinite Dimensional Topology. Takayuki Kihara
Recursion Theoretic Methods in Descriptive Set Theory and Infinite Dimensional Topology Takayuki Kihara Department of Mathematics, University of California, Berkeley, USA Joint Work with Arno Pauly (University
More informationSolutions to Tutorial 8 (Week 9)
The University of Sydney School of Mathematics and Statistics Solutions to Tutorial 8 (Week 9) MATH3961: Metric Spaces (Advanced) Semester 1, 2018 Web Page: http://www.maths.usyd.edu.au/u/ug/sm/math3961/
More informationSubstrictly Cyclic Operators
Substrictly Cyclic Operators Ben Mathes dbmathes@colby.edu April 29, 2008 Dedicated to Don Hadwin Abstract We initiate the study of substrictly cyclic operators and algebras. As an application of this
More informationReal Analysis Chapter 4 Solutions Jonathan Conder
2. Let x, y X and suppose that x y. Then {x} c is open in the cofinite topology and contains y but not x. The cofinite topology on X is therefore T 1. Since X is infinite it contains two distinct points
More informationMTG 5316/4302 FALL 2018 REVIEW FINAL
MTG 5316/4302 FALL 2018 REVIEW FINAL JAMES KEESLING Problem 1. Define open set in a metric space X. Define what it means for a set A X to be connected in a metric space X. Problem 2. Show that if a set
More informationLebesgue Measure on R n
CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical sets
More informationSOME ADDITIVE DARBOUX LIKE FUNCTIONS
Journal of Applied Analysis Vol. 4, No. 1 (1998), pp. 43 51 SOME ADDITIVE DARBOUX LIKE FUNCTIONS K. CIESIELSKI Received June 11, 1997 and, in revised form, September 16, 1997 Abstract. In this note we
More informationThe ideal of Sierpiński-Zygmund sets on the plane
The ideal of Sierpiński-Zygmund sets on the plane Krzysztof P lotka Department of Mathematics, West Virginia University Morgantown, WV 26506-6310, USA kplotka@math.wvu.edu and Institute of Mathematics,
More informationUniquely Universal Sets
Uniquely Universal Sets 1 Uniquely Universal Sets Abstract 1 Arnold W. Miller We say that X Y satisfies the Uniquely Universal property (UU) iff there exists an open set U X Y such that for every open
More informationOPERATORS ON TWO BANACH SPACES OF CONTINUOUS FUNCTIONS ON LOCALLY COMPACT SPACES OF ORDINALS
OPERATORS ON TWO BANACH SPACES OF CONTINUOUS FUNCTIONS ON LOCALLY COMPACT SPACES OF ORDINALS TOMASZ KANIA AND NIELS JAKOB LAUSTSEN Abstract Denote by [0, ω 1 ) the set of countable ordinals, equipped with
More informationON SPACES IN WHICH COMPACT-LIKE SETS ARE CLOSED, AND RELATED SPACES
Commun. Korean Math. Soc. 22 (2007), No. 2, pp. 297 303 ON SPACES IN WHICH COMPACT-LIKE SETS ARE CLOSED, AND RELATED SPACES Woo Chorl Hong Reprinted from the Communications of the Korean Mathematical Society
More informationThe Polynomial Numerical Index of L p (µ)
KYUNGPOOK Math. J. 53(2013), 117-124 http://dx.doi.org/10.5666/kmj.2013.53.1.117 The Polynomial Numerical Index of L p (µ) Sung Guen Kim Department of Mathematics, Kyungpook National University, Daegu
More information1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3
Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,
More informationChapter 3: Baire category and open mapping theorems
MA3421 2016 17 Chapter 3: Baire category and open mapping theorems A number of the major results rely on completeness via the Baire category theorem. 3.1 The Baire category theorem 3.1.1 Definition. A
More informationTopological groups with dense compactly generated subgroups
Applied General Topology c Universidad Politécnica de Valencia Volume 3, No. 1, 2002 pp. 85 89 Topological groups with dense compactly generated subgroups Hiroshi Fujita and Dmitri Shakhmatov Abstract.
More informationFunctional Analysis HW #1
Functional Analysis HW #1 Sangchul Lee October 9, 2015 1 Solutions Solution of #1.1. Suppose that X
More informationThe projectivity of C -algebras and the topology of their spectra
The projectivity of C -algebras and the topology of their spectra Zinaida Lykova Newcastle University, UK Waterloo 2011 Typeset by FoilTEX 1 The Lifting Problem Let A be a Banach algebra and let A-mod
More informationREAL AND COMPLEX ANALYSIS
REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any
More informationAxioms of separation
Axioms of separation These notes discuss the same topic as Sections 31, 32, 33, 34, 35, and also 7, 10 of Munkres book. Some notions (hereditarily normal, perfectly normal, collectionwise normal, monotonically
More informationint cl int cl A = int cl A.
BAIRE CATEGORY CHRISTIAN ROSENDAL 1. THE BAIRE CATEGORY THEOREM Theorem 1 (The Baire category theorem. Let (D n n N be a countable family of dense open subsets of a Polish space X. Then n N D n is dense
More informationA NEW LINDELOF SPACE WITH POINTS G δ
A NEW LINDELOF SPACE WITH POINTS G δ ALAN DOW Abstract. We prove that implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality 2 ℵ1 which has points G δ. In addition, this space has
More informationOn minimal models of the Region Connection Calculus
Fundamenta Informaticae 69 (2006) 1 20 1 IOS Press On minimal models of the Region Connection Calculus Lirong Xia State Key Laboratory of Intelligent Technology and Systems Department of Computer Science
More informationChapter 2 Metric Spaces
Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics
More informationALGEBRAIC SUMS OF SETS IN MARCZEWSKI BURSTIN ALGEBRAS
François G. Dorais, Department of Mathematics, Dartmouth College, 6188 Bradley Hall, Hanover, NH 03755, USA (e-mail: francois.g.dorais@dartmouth.edu) Rafa l Filipów, Institute of Mathematics, University
More informationSome Background Material
Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important
More informationarxiv: v1 [math.fa] 14 May 2008
DEFINABILITY UNDER DUALITY arxiv:0805.2036v1 [math.fa] 14 May 2008 PANDELIS DODOS Abstract. It is shown that if A is an analytic class of separable Banach spaces with separable dual, then the set A = {Y
More informationStone-Čech compactification of Tychonoff spaces
The Stone-Čech compactification of Tychonoff spaces Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto June 27, 2014 1 Completely regular spaces and Tychonoff spaces A topological
More informationGENERALIZED CANTOR SETS AND SETS OF SUMS OF CONVERGENT ALTERNATING SERIES
Journal of Applied Analysis Vol. 7, No. 1 (2001), pp. 131 150 GENERALIZED CANTOR SETS AND SETS OF SUMS OF CONVERGENT ALTERNATING SERIES M. DINDOŠ Received September 7, 2000 and, in revised form, February
More informationINSERTION OF A FUNCTION BELONGING TO A CERTAIN SUBCLASS OF R X
Bulletin of the Iranian Mathematical Society, Vol. 28 No. 2 (2002), pp.19-27 INSERTION OF A FUNCTION BELONGING TO A CERTAIN SUBCLASS OF R X MAJID MIRMIRAN Abstract. Let X be a topological space and E(X,
More informationHomeomorphism groups of Sierpiński carpets and Erdős space
F U N D A M E N T A MATHEMATICAE 207 (2010) Homeomorphism groups of Sierpiński carpets and Erdős space by Jan J. Dijkstra and Dave Visser (Amsterdam) Abstract. Erdős space E is the rational Hilbert space,
More informationMAS3706 Topology. Revision Lectures, May I do not answer enquiries as to what material will be in the exam.
MAS3706 Topology Revision Lectures, May 208 Z.A.Lykova It is essential that you read and try to understand the lecture notes from the beginning to the end. Many questions from the exam paper will be similar
More informationLecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University
Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................
More informationNotes for Functional Analysis
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
More informationP-adic Functions - Part 1
P-adic Functions - Part 1 Nicolae Ciocan 22.11.2011 1 Locally constant functions Motivation: Another big difference between p-adic analysis and real analysis is the existence of nontrivial locally constant
More informationarxiv:math/ v1 [math.fa] 26 Oct 1993
arxiv:math/9310217v1 [math.fa] 26 Oct 1993 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M.I.Ostrovskii Abstract. It is proved that there exist complemented subspaces of countable topological
More informationTHE GEOMETRY OF CONVEX TRANSITIVE BANACH SPACES
THE GEOMETRY OF CONVEX TRANSITIVE BANACH SPACES JULIO BECERRA GUERRERO AND ANGEL RODRIGUEZ PALACIOS 1. Introduction Throughout this paper, X will denote a Banach space, S S(X) and B B(X) will be the unit
More informationz -FILTERS AND RELATED IDEALS IN C(X) Communicated by B. Davvaz
Algebraic Structures and Their Applications Vol. 2 No. 2 ( 2015 ), pp 57-66. z -FILTERS AND RELATED IDEALS IN C(X) R. MOHAMADIAN Communicated by B. Davvaz Abstract. In this article we introduce the concept
More informationTopology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124
More informationCHAPTER X THE SPECTRAL THEOREM OF GELFAND
CHAPTER X THE SPECTRAL THEOREM OF GELFAND DEFINITION A Banach algebra is a complex Banach space A on which there is defined an associative multiplication for which: (1) x (y + z) = x y + x z and (y + z)
More informationGELFAND S THEOREM. Christopher McMurdie. Advisor: Arlo Caine. Spring 2004
GELFAND S THEOREM Christopher McMurdie Advisor: Arlo Caine Super Advisor: Dr Doug Pickrell Spring 004 This paper is a proof of the first part of Gelfand s Theorem, which states the following: Every compact,
More informationIsometries of spaces of unbounded continuous functions
Isometries of spaces of unbounded continuous functions Jesús Araujo University of Cantabria, Spain Krzysztof Jarosz Southern Illinois University at Edwardsville, USA Abstract. By the classical Banach-Stone
More informationDEFINABILITY UNDER DUALITY. 1. Introduction
DEFINABILITY UNDER DUALITY PANDELIS DODOS Abstract. It is shown that if A is an analytic class of separable Banach spaces with separable dual, then the set A = {Y : X A with Y = X } is analytic. The corresponding
More informationON DENSITY TOPOLOGIES WITH RESPECT
Journal of Applied Analysis Vol. 8, No. 2 (2002), pp. 201 219 ON DENSITY TOPOLOGIES WITH RESPECT TO INVARIANT σ-ideals J. HEJDUK Received June 13, 2001 and, in revised form, December 17, 2001 Abstract.
More informationDecomposition of Riesz frames and wavelets into a finite union of linearly independent sets
Decomposition of Riesz frames and wavelets into a finite union of linearly independent sets Ole Christensen, Alexander M. Lindner Abstract We characterize Riesz frames and prove that every Riesz frame
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week 2 November 6 November Deadline to hand in the homeworks: your exercise class on week 9 November 13 November Exercises (1) Let X be the following space of piecewise
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationON C*-ALGEBRAS WHICH CANNOT BE DECOMPOSED INTO TENSOR PRODUCTS WITH BOTH FACTORS INFINITE-DIMENSIONAL
ON C*-ALGEBRAS WHICH CANNOT BE DECOMPOSED INTO TENSOR PRODUCTS WITH BOTH FACTORS INFINITE-DIMENSIONAL TOMASZ KANIA Abstract. We prove that C*-algebras which satisfy a certain Banach-space property cannot
More informationTHE NEARLY ADDITIVE MAPS
Bull. Korean Math. Soc. 46 (009), No., pp. 199 07 DOI 10.4134/BKMS.009.46..199 THE NEARLY ADDITIVE MAPS Esmaeeil Ansari-Piri and Nasrin Eghbali Abstract. This note is a verification on the relations between
More information2.4 Annihilators, Complemented Subspaces
40 CHAPTER 2. WEAK TOPOLOGIES AND REFLEXIVITY 2.4 Annihilators, Complemented Subspaces Definition 2.4.1. (Annihilators, Pre-Annihilators) Assume X is a Banach space. Let M X and N X. We call the annihilator
More informationThe structure of ideals, point derivations, amenability and weak amenability of extended Lipschitz algebras
Int. J. Nonlinear Anal. Appl. 8 (2017) No. 1, 389-404 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2016.493 The structure of ideals, point derivations, amenability and weak amenability
More informationPROBLEMS. (b) (Polarization Identity) Show that in any inner product space
1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization
More informationQuasi-invariant measures for continuous group actions
Contemporary Mathematics Quasi-invariant measures for continuous group actions Alexander S. Kechris Dedicated to Simon Thomas on his 60th birthday Abstract. The class of ergodic, invariant probability
More informationconsists of two disjoint copies of X n, each scaled down by 1,
Homework 4 Solutions, Real Analysis I, Fall, 200. (4) Let be a topological space and M be a σ-algebra on which contains all Borel sets. Let m, µ be two positive measures on M. Assume there is a constant
More informationDENSE C-EMBEDDED SUBSPACES OF PRODUCTS
DENSE C-EMBEDDED SUBSPACES OF PRODUCTS W. W. COMFORT, IVAN GOTCHEV, AND LUIS RECODER-NÚÑEZ Abstract. Throughout this paper the spaces X i are assumed Tychonoff, (X I ) κ denotes X I := Π i I X i with the
More informationMath 209B Homework 2
Math 29B Homework 2 Edward Burkard Note: All vector spaces are over the field F = R or C 4.6. Two Compactness Theorems. 4. Point Set Topology Exercise 6 The product of countably many sequentally compact
More informationA CHARACTERIZATION OF POWER HOMOGENEITY G. J. RIDDERBOS
A CHARACTERIZATION OF POWER HOMOGENEITY G. J. RIDDERBOS Abstract. We prove that every -power homogeneous space is power homogeneous. This answers a question of the author and it provides a characterization
More informationMath 676. A compactness theorem for the idele group. and by the product formula it lies in the kernel (A K )1 of the continuous idelic norm
Math 676. A compactness theorem for the idele group 1. Introduction Let K be a global field, so K is naturally a discrete subgroup of the idele group A K and by the product formula it lies in the kernel
More informationA Countably-Based Domain Representation of a Non-Regular Hausdorff Space
A Countably-Based Domain Representation of a Non-Regular Hausdorff Space Harold Bennett and David Lutzer April 4, 2006 Abstract In this paper we give an example of a countably-based algebraic domain D
More informationREVIEW OF ESSENTIAL MATH 346 TOPICS
REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations
More informationB 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.
Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2
More information