A NOTE ON LEBESGUE SPACES
|
|
- Berniece Marybeth Franklin
- 5 years ago
- Views:
Transcription
1 Volume 6, 1981 Pages A NOTE ON LEBESGUE SPACES by Sam B. Nadler, Jr. ad Thelma West Topology Proceedigs Web: Mail: Topology Proceedigs Departmet of Mathematics & Statistics Aubur Uiversity, Alabama 36849, USA topolog@aubur.edu ISSN: COPYRIGHT c by Topology Proceedigs. All rights reserved.
2 TOPOLOGY PROCEEDINGS Volume A NOTE ON LEBESGUE SPACES Sam B. Nadler, Jr. ad Thelma West L~t (X,d) be a metric space. A Lebesgue umber for a ope cover lj of X is a E > 0 such that for each poit p E X, the ope,ball Bd(p,E) = {x E X: d(p,x) < E} is cotaied i at least oe member of 0. A Lebesgue space (L-space)(is a metric space such that every ope cover of the space has a Lebesgue umber. It is kow that the L-spaces are precisely those metric spaces for which every cotiuous real-valued fuctio is uiformly cotiuous ([6, p. 112], [1, p. 12]). I particular, every compact metric space is a L-space. We will show that there are L-spaces which are ot eve locally compact. Furthermore, i Theorem 1 we characterize those metric spaces which cotai olocally compact L-subspaces. The proof of Theorem 1 shows how to costruct such subspaces i the most geeral possible settig. I Theorem 2 we show that ay L-space must belocally compact "at most poits. II I Theorem 3 we obtai a simple ecessary ad sufficiet coditio i order that a metric space be a locally compact L-space. This coditio determies the structure of all such spaces. We will briefly discuss Theorem 3 i r~latio to results i [4] ad [S]--see the Remark at the ed of the paper. Recall tha~ a metric space (X,d) is said to be totally bouded provided that for each E > 0, there exist fiitely may poits Pl,P2,,P of X such that X U{Bd(Pi,E): i = 1,2,,} [2, p. 22]. A subset A of a metric space
3 364 Nadler ad West (X,d) is said to be uifopmly isolated provided that there exists a C > 0 such that d(x,x') ~ c for all x,x' E A such that x ~ x' [5, p. 153]. Clearly, a metric space fails to be totally bouded if ad oly if it cotais a ifiite uiformly isolated subset. For use later o, let us ote the followig lemma. It follows easily from Theorem 1 of [1]. Lemma 1. A metric space (X,d) is a L-space if ad oly if the set L of all limit poits of X is compact ad for each ope subset U of X such that U.~ L, X - U is uifopmly isolated. The followig theorem is our characterizatio of those metric spaces which cotai olocally compact L-subspaces. Theorem 1. Let (M,D) be a metpic space. The: Every L-subspace of M is locally compact if ad oly if every poit of M has a totally-bouded eighbophood, i.e., if ad oly if the completio of M is locally compact. Ppoof. Assume that there is a poit p E M such that o eighborhood of p is totally bouded. The, as oted above, each eighborhood of p cotais-a ifiite uriiformly isolated subset. Let Xl be a ifiite uiformly isolated subset of the ball B o (p,2-1 ) such that p Xl. Assume iductively that we have chose ifiite subsets Xi of B O (p,2 -i ) - {p} for each i = 1,2,., ( < 00) such that Ui=IX i is uiformly isolated. Let o = if{d(p,x): x E ui=ix i } ad ote that c > O. Let E mi{c/2,2- - I }. Sice E > 0,
4 TOPOLOGY PROCEEDINGS Volume there is a ifiite uiformly isolated subset X 1 of + BD(p, ) such that p X + l. Note that u~:txi is uiformly isolated. Thus, we have iductively defied X for each = 1,2,--- such that, for each k = 1,2,---, uk X is ui=l formly isolated. Let X = {p} U (U:=lX ) ad let d deote the subspace metric for X obtaied from D. We see that p is the oly limit poit of X ad that if U is k ay ope subset of X such that p E U, the X - U c U=lX for some k < 00 ad, thus, X - U is uiformly isolated. Hece, by Lemma 1, (X,d) is a L-space. Furthermore, as is easy to see, (X,d) is ot locally compact at p. Therefore, we have proved half of Theorem 1. To prove the other half, assume that every poit of M has a totally bouded eighborhood ad let Y be a L-subspace of M. It follows easily from Lemma 1 that Y is complete (the fact that L-spaces are complete is also oted i [6, 8, p. 112]). Let y E Y. From our assumptio about M, there is a totally bouded eighborhood N(y) of y. We assume without loss of geerality that N(y) is a closed subset of M. The, N(y) Y is a closed eighborhood of y i Y. Sice Y is complete, N(y) Y is complete. Thus, sice N(y) Y is totally bouded, N(y) Y is compact [2, p. 22]. Hece; we have proved that each poit of Y has a compact eighborhood i Y. Therefore, Y is locally compact. This completes the proof of Theorem 1. For metric liear topological vector spaces, Theorem 1. becomes the followig result:
5 366 Nadler ad West Corlllary. A metric liear topological vector space is fiite dimesioal if ad ~ly if every subset which is a L-space is locally compact. Proof. The corollary follows immediately from Th~orem 1 above ad 7.8 of [3, p. 62]. The L-spaces costructed i the proof of Theorem 1 fail to be locally compact at oly oe poit. The questio arises as to whether there is a L-space which fails to be locally compact at every poit of some oempty ope subset. The followig result aswers this questio by showig that o such L-space exists. Theorem 2. If (X,d) is a L-space~ the X is locally compact at each poit of a dese ope subset of x. Proof. Let L = {x: x is a limit poit of X} ad let W = {x X: X is locally compact at x}. Note that X - W c L. Thus, sice L is compact (by Lemma 1), X - W ca ot cotai ay oempty ope subset of X. Hece, W is a dese subset of X. Clearly, W is a ope subset of X. This completes the proof of Theorem 2. We have the followig characterizatio of locally compact L-spaces: Theorem 3. A metric space (X,d) is a locally compact L-space if ad oly if X is the uio of a compact subspace ad a uiformly isolated subspace. Proof. Assume that (X,d) is a locally compact L-space. Let L deote the set of all limit poits of X. By Lemma 1,
6 TOPOLOGY PROCEEDINGS Volume L is compact. Thus, sice X is locally compact, L ca be covered by fiitely may ope subsets of X whose closures are compact. Hece, there is a ope subset U of X such that LeU ad such that the closure IT of U is compact. By Lemma 1, X - U is uiformly isolated. Therefore, writig X = IT u (X - U), we see that we have proved half of Theorem 3. Now, to prove the other half of Theorem 3, assume that (X,d) is a metric space such that X is the uio of a compact subspace C ad a uiformly isolated subspace E. We first show that X is locally compact. Note that E - C is a uiformly isolated ope subspace of X. Thus, for each poit x E E - C, we see that {x} is a ope subset of X. Hece, clearly, X is locally compact at each poit of E - C. To show that X is locally compact at each poit of C, it suffices (sice C is compact) to show that C is a ope subset of X. Suppose that C is ot a ope subset of X. The there is a sequece {x}~=l of poits of X - C such that {x}~=l coverges to a poit of C. Sice {x}~=l is a Cauchy sequece of poits of E, we have a cotradictio to the assumptio that E is uiformly isolated. Thus, C is a ope subspace of X. This completes the proof that X is locally compact. Next we show usig Lemma 1 that (X,d) is a L-space. Let L deote the set of all limit poits of X. It was show above that for each x E E - C, {x} is a ope subset of X. Hece, L ~ C. Thus, sice L is a closed subset of X ad sice C is compact, we have that L is compact. Let U be a ope subset of X such that U ~ L. We wish to show that X - Uis uiformly isolated (see Lemma 1). Note that
7 368 Nadler ad West (*) X - U = (C - U) U (E - U) Sice C - U is a closed subset of the compact set C, C - U is compact. Sice U ~ L, C - U cotais o limit poit of X. Thus, C - U must be fiite. Sice E - U is a subset of the uiformly isolated space E, E - U is uiformly isolated. It ow follows from (*) that X - U is uiformly isolated. Therefore, we ca ow use Lemma 1 to coclude that (X,d) is a L-space. This completes the proof of Theorem 3. Remark. Let E be a subset of a metric space (M,d). Cosider the followig two coditios o E: (1) every cotiuous real-valued fuctio o E is uiformly cotiuous; isolated. (2) E = E U E 2 where E is compact ad E 2 is uiformly l l Recall from the begiig of this paper that (1) is equiva let to E beig a L-space. (M,d) is locally compact. Assume that (1) holds ad that Sice E is complete [5, Thro. 1], E is a closed, therefore locally compact subspace of M. Hece, by Theorem 3 above, (2) holds. With o restrictio o (M,d), we see from Theorem 3 that (2) implies (1). I Theorem 4 of [5] it is show that (1) implies (2) if closed ad bouded subsets of (M,d) are compact. However, as we have just show, (1) is equivalet to (2) i more geeral spaces (M,d). Similar geeralizatios of some other results i [5] are also possible to obtai by usig our results. I coectio with this we ote that coditio (2), which occurs frequetly i results i [4] ad [5], is really equivalet to E beig a locally compact L-space (by our Theorem 3).
8 TOPOLOGY PROCEEDINGS Volume The authors are grateful to Professor Norma Levie for refereces [1] ad [6]. Refereces 1. M. Atsuji, Uiform cotiuity of cotiuous fuctios of metric spaces, Pac. J. Math. a (1958), N. Duford ad J. T. Schwartz, Liear operato~s: Part I, Itersciece Publishers, New York, J. L. Kelley ad I. Namioka, Liear topological spaces, D. Va Nostrad Compay, Ic., Priceto, New Jersey, N. Levie, Uiformly cotiuous liear sets, Amer. Math. Mothly 62 (1955), ad W. G. Sauders, Uiformly cotiuous sets i metric spaces, Amer. Math. Mothly 67 (1960), A. A. Moteiro ad M. M. Peixoto, Le ombre de Lebesgue et la cotiuite uiforme, Portuga1iae Mathematica 10 (1951), West Virgiia Uiversity Morgatow, West Virgiia 26506
Theorem 3. A subset S of a topological space X is compact if and only if every open cover of S by open sets in X has a finite subcover.
Compactess Defiitio 1. A cover or a coverig of a topological space X is a family C of subsets of X whose uio is X. A subcover of a cover C is a subfamily of C which is a cover of X. A ope cover of X is
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationOn Strictly Point T -asymmetric Continua
Volume 35, 2010 Pages 91 96 http://topology.aubur.edu/tp/ O Strictly Poit T -asymmetric Cotiua by Leobardo Ferádez Electroically published o Jue 19, 2009 Topology Proceedigs Web: http://topology.aubur.edu/tp/
More informationLecture Notes for Analysis Class
Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationA NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS
Acta Math. Hugar., 2007 DOI: 10.1007/s10474-007-7013-6 A NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS L. L. STACHÓ ad L. I. SZABÓ Bolyai Istitute, Uiversity of Szeged, Aradi vértaúk tere 1, H-6720
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationSOME REMARKS ON FREELY DECOMPOSABLE MAPPINGS
Volume, 1977 Pages 13 17 http://topology.aubur.edu/tp/ SOME REMARKS ON FREELY DECOMPOSABLE MAPPINGS by C. Bruce Hughes Topology Proceedigs Web: http://topology.aubur.edu/tp/ Mail: Topology Proceedigs Departmet
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More informationMath 341 Lecture #31 6.5: Power Series
Math 341 Lecture #31 6.5: Power Series We ow tur our attetio to a particular kid of series of fuctios, amely, power series, f(x = a x = a 0 + a 1 x + a 2 x 2 + where a R for all N. I terms of a series
More informationON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX0000-0 ON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS MARCH T. BOEDIHARDJO AND WILLIAM B. JOHNSON 2
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationMetric Space Properties
Metric Space Properties Math 40 Fial Project Preseted by: Michael Brow, Alex Cordova, ad Alyssa Sachez We have already poited out ad will recogize throughout this book the importace of compact sets. All
More informationMATH 413 FINAL EXAM. f(x) f(y) M x y. x + 1 n
MATH 43 FINAL EXAM Math 43 fial exam, 3 May 28. The exam starts at 9: am ad you have 5 miutes. No textbooks or calculators may be used durig the exam. This exam is prited o both sides of the paper. Good
More informationTopologie. Musterlösungen
Fakultät für Mathematik Sommersemester 2018 Marius Hiemisch Topologie Musterlösuge Aufgabe (Beispiel 1.2.h aus Vorlesug). Es sei X eie Mege ud R Abb(X, R) eie Uteralgebra, d.h. {kostate Abbilduge} R ud
More informationIntegrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number
MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios
More informationMORE ON CAUCHY CONDITIONS
Volume 9, 1984 Pages 31 36 http://topology.aubur.edu/tp/ MORE ON CAUCHY CONDITIONS by S. W. Davis Topology Proceedigs Web: http://topology.aubur.edu/tp/ Mail: Topology Proceedigs Departmet of Mathematics
More informationREAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS
REAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS 18th Feb, 016 Defiitio (Lipschitz fuctio). A fuctio f : R R is said to be Lipschitz if there exists a positive real umber c such that for ay x, y i the domai
More informationA NOTE ON SPECTRAL CONTINUITY. In Ho Jeon and In Hyoun Kim
Korea J. Math. 23 (2015), No. 4, pp. 601 605 http://dx.doi.org/10.11568/kjm.2015.23.4.601 A NOTE ON SPECTRAL CONTINUITY I Ho Jeo ad I Hyou Kim Abstract. I the preset ote, provided T L (H ) is biquasitriagular
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationBoundaries and the James theorem
Boudaries ad the James theorem L. Vesely 1. Itroductio The followig theorem is importat ad well kow. All spaces cosidered here are real ormed or Baach spaces. Give a ormed space X, we deote by B X ad S
More informationf n (x) f m (x) < ɛ/3 for all x A. By continuity of f n and f m we can find δ > 0 such that d(x, x 0 ) < δ implies that
Lecture 15 We have see that a sequece of cotiuous fuctios which is uiformly coverget produces a limit fuctio which is also cotiuous. We shall stregthe this result ow. Theorem 1 Let f : X R or (C) be a
More informationANSWERS TO MIDTERM EXAM # 2
MATH 03, FALL 003 ANSWERS TO MIDTERM EXAM # PENN STATE UNIVERSITY Problem 1 (18 pts). State ad prove the Itermediate Value Theorem. Solutio See class otes or Theorem 5.6.1 from our textbook. Problem (18
More informationA REMARK ON A PROBLEM OF KLEE
C O L L O Q U I U M M A T H E M A T I C U M VOL. 71 1996 NO. 1 A REMARK ON A PROBLEM OF KLEE BY N. J. K A L T O N (COLUMBIA, MISSOURI) AND N. T. P E C K (URBANA, ILLINOIS) This paper treats a property
More informationHOMEWORK #4 - MA 504
HOMEWORK #4 - MA 504 PAULINHO TCHATCHATCHA Chapter 2, problem 19. (a) If A ad B are disjoit closed sets i some metric space X, prove that they are separated. (b) Prove the same for disjoit ope set. (c)
More informationTHE CANTOR INTERMEDIATE VALUE PROPERTY
Volume 7, 198 Pages 55 6 http://topology.aubur.edu/tp/ THE CANTOR INTERMEDIATE VALUE PROPERTY by Richard G. Gibso ad Fred Roush Topology Proceedigs Web: http://topology.aubur.edu/tp/ Mail: Topology Proceedigs
More informationMeasure and Measurable Functions
3 Measure ad Measurable Fuctios 3.1 Measure o a Arbitrary σ-algebra Recall from Chapter 2 that the set M of all Lebesgue measurable sets has the followig properties: R M, E M implies E c M, E M for N implies
More informationFUNDAMENTALS OF REAL ANALYSIS by
FUNDAMENTALS OF REAL ANALYSIS by Doğa Çömez Backgroud: All of Math 450/1 material. Namely: basic set theory, relatios ad PMI, structure of N, Z, Q ad R, basic properties of (cotiuous ad differetiable)
More informationTREE-LIKE CONTINUA AS LIMITS OF CYCLIC GRAPHS
Volume 4, 1979 Pages 507 514 http://topology.aubur.edu/tp/ TREE-LIKE CONTINUA AS LIMITS OF CYCLIC GRAPHS by Lex G. Oversteege ad James T. Rogers, Jr. Topology Proceedigs Web: http://topology.aubur.edu/tp/
More informationMath 140A Elementary Analysis Homework Questions 3-1
Math 0A Elemetary Aalysis Homework Questios -.9 Limits Theorems for Sequeces Suppose that lim x =, lim y = 7 ad that all y are o-zero. Detarime the followig limits: (a) lim(x + y ) (b) lim y x y Let s
More informationSolution. 1 Solutions of Homework 1. Sangchul Lee. October 27, Problem 1.1
Solutio Sagchul Lee October 7, 017 1 Solutios of Homework 1 Problem 1.1 Let Ω,F,P) be a probability space. Show that if {A : N} F such that A := lim A exists, the PA) = lim PA ). Proof. Usig the cotiuity
More informationEquivalent Banach Operator Ideal Norms 1
It. Joural of Math. Aalysis, Vol. 6, 2012, o. 1, 19-27 Equivalet Baach Operator Ideal Norms 1 Musudi Sammy Chuka Uiversity College P.O. Box 109-60400, Keya sammusudi@yahoo.com Shem Aywa Maside Muliro Uiversity
More informationUniversity of Colorado Denver Dept. Math. & Stat. Sciences Applied Analysis Preliminary Exam 13 January 2012, 10:00 am 2:00 pm. Good luck!
Uiversity of Colorado Dever Dept. Math. & Stat. Scieces Applied Aalysis Prelimiary Exam 13 Jauary 01, 10:00 am :00 pm Name: The proctor will let you read the followig coditios before the exam begis, ad
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationProperties of Fuzzy Length on Fuzzy Set
Ope Access Library Joural 206, Volume 3, e3068 ISSN Olie: 2333-972 ISSN Prit: 2333-9705 Properties of Fuzzy Legth o Fuzzy Set Jehad R Kider, Jaafar Imra Mousa Departmet of Mathematics ad Computer Applicatios,
More informationCouncil for Innovative Research
ABSTRACT ON ABEL CONVERGENT SERIES OF FUNCTIONS ERDAL GÜL AND MEHMET ALBAYRAK Yildiz Techical Uiversity, Departmet of Mathematics, 34210 Eseler, Istabul egul34@gmail.com mehmetalbayrak12@gmail.com I this
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationBETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS. 1. Introduction. Throughout the paper we denote by X a linear space and by Y a topological linear
BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS Abstract. The aim of this paper is to give sufficiet coditios for a quasicovex setvalued mappig to be covex. I particular, we recover several kow characterizatios
More information5 Many points of continuity
Tel Aviv Uiversity, 2013 Measure ad category 40 5 May poits of cotiuity 5a Discotiuous derivatives.............. 40 5b Baire class 1 (classical)............... 42 5c Baire class 1 (moder)...............
More informationEXAMPLES OF HEREDITARILY STRONGLY INFINITE-DIMENSIONAL COMPACTA
Volume 3, 1978 Pages 495 506 http://topology.aubur.edu/tp/ EXAMPLES OF HEREDITARILY STRONGLY INFINITE-DIMENSIONAL COMPACTA by R. M. Schori ad Joh J. Walsh Topology Proceedigs Web: http://topology.aubur.edu/tp/
More informationMATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and
MATH01 Real Aalysis (2008 Fall) Tutorial Note #7 Sequece ad Series of fuctio 1: Poitwise Covergece ad Uiform Covergece Part I: Poitwise Covergece Defiitio of poitwise covergece: A sequece of fuctios f
More informationMath 203A, Solution Set 8.
Math 20A, Solutio Set 8 Problem 1 Give four geeral lies i P, show that there are exactly 2 lies which itersect all four of them Aswer: Recall that the space of lies i P is parametrized by the Grassmaia
More informationThe Borel hierarchy classifies subsets of the reals by their topological complexity. Another approach is to classify them by size.
Lecture 7: Measure ad Category The Borel hierarchy classifies subsets of the reals by their topological complexity. Aother approach is to classify them by size. Filters ad Ideals The most commo measure
More informationMTG 6316 HOMEWORK Spring 2017
MTG 636 HOMEWORK Sprig 207 53. Let {U k } k= be a fiite ope cover of X ad f k : U k! Y be cotiuous for each k =,...,. Show that if f k (x) = f j (x) for all x 2 U k \ U j, the the fuctio F : X! Y defied
More informationMath 132, Fall 2009 Exam 2: Solutions
Math 3, Fall 009 Exam : Solutios () a) ( poits) Determie for which positive real umbers p, is the followig improper itegral coverget, ad for which it is diverget. Evaluate the itegral for each value of
More information1 Lecture 2: Sequence, Series and power series (8/14/2012)
Summer Jump-Start Program for Aalysis, 202 Sog-Yig Li Lecture 2: Sequece, Series ad power series (8/4/202). More o sequeces Example.. Let {x } ad {y } be two bouded sequeces. Show lim sup (x + y ) lim
More informationMcGill University Math 354: Honors Analysis 3 Fall 2012 Solutions to selected problems
McGill Uiversity Math 354: Hoors Aalysis 3 Fall 212 Assigmet 3 Solutios to selected problems Problem 1. Lipschitz fuctios. Let Lip K be the set of all fuctios cotiuous fuctios o [, 1] satisfyig a Lipschitz
More informationIt is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.
Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable
More informationLecture 2. The Lovász Local Lemma
Staford Uiversity Sprig 208 Math 233A: No-costructive methods i combiatorics Istructor: Ja Vodrák Lecture date: Jauary 0, 208 Origial scribe: Apoorva Khare Lecture 2. The Lovász Local Lemma 2. Itroductio
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationFall 2013 MTH431/531 Real analysis Section Notes
Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationArchimedes - numbers for counting, otherwise lengths, areas, etc. Kepler - geometry for planetary motion
Topics i Aalysis 3460:589 Summer 007 Itroductio Ree descartes - aalysis (breaig dow) ad sythesis Sciece as models of ature : explaatory, parsimoious, predictive Most predictios require umerical values,
More informationBeyond simple iteration of a single function, or even a finite sequence of functions, results
A Primer o the Elemetary Theory of Ifiite Compositios of Complex Fuctios Joh Gill Sprig 07 Abstract: Elemetary meas ot requirig the complex fuctios be holomorphic Theorem proofs are fairly simple ad are
More informationCommon Coupled Fixed Point of Mappings Satisfying Rational Inequalities in Ordered Complex Valued Generalized Metric Spaces
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn:319-765x Volume 10, Issue 3 Ver II (May-Ju 014), PP 69-77 Commo Coupled Fixed Poit of Mappigs Satisfyig Ratioal Iequalities i Ordered Complex
More informationFilippov Implicit Function Theorem for Quasi-Caratheodory Functions
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 214, 475481 1997 ARTICLE NO. AY975584 Filippov Implicit Fuctio Theorem for Quasi-Caratheodory Fuctios M. Didos ˇ ad V. Toma Faculty of Mathematics ad Physics,
More informationSEMIGROUPS OF VALUATIONS DOMINATING LOCAL DOMAINS
SEMIGROUPS OF VALUATIONS DOMINATING LOCAL DOMAINS STEVEN DALE CUTKOSKY Let (R, m R ) be a equicharacteristic local domai, with quotiet field K. Suppose that ν is a valuatio of K with valuatio rig (V, m
More informationA constructive analysis of convex-valued demand correspondence for weakly uniformly rotund and monotonic preference
MPRA Muich Persoal RePEc Archive A costructive aalysis of covex-valued demad correspodece for weakly uiformly rotud ad mootoic preferece Yasuhito Taaka ad Atsuhiro Satoh. May 04 Olie at http://mpra.ub.ui-mueche.de/55889/
More informationMATH4822E FOURIER ANALYSIS AND ITS APPLICATIONS
MATH48E FOURIER ANALYSIS AND ITS APPLICATIONS 7.. Cesàro summability. 7. Summability methods Arithmetic meas. The followig idea is due to the Italia geometer Eresto Cesàro (859-96). He shows that eve if
More informationFinal Solutions. 1. (25pts) Define the following terms. Be as precise as you can.
Mathematics H104 A. Ogus Fall, 004 Fial Solutios 1. (5ts) Defie the followig terms. Be as recise as you ca. (a) (3ts) A ucoutable set. A ucoutable set is a set which ca ot be ut ito bijectio with a fiite
More informationThe value of Banach limits on a certain sequence of all rational numbers in the interval (0,1) Bao Qi Feng
The value of Baach limits o a certai sequece of all ratioal umbers i the iterval 0, Bao Qi Feg Departmet of Mathematical Scieces, Ket State Uiversity, Tuscarawas, 330 Uiversity Dr. NE, New Philadelphia,
More information6. Uniform distribution mod 1
6. Uiform distributio mod 1 6.1 Uiform distributio ad Weyl s criterio Let x be a seuece of real umbers. We may decompose x as the sum of its iteger part [x ] = sup{m Z m x } (i.e. the largest iteger which
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
More informationpage Suppose that S 0, 1 1, 2.
page 10 1. Suppose that S 0, 1 1,. a. What is the set of iterior poits of S? The set of iterior poits of S is 0, 1 1,. b. Give that U is the set of iterior poits of S, evaluate U. 0, 1 1, 0, 1 1, S. The
More informationf(x)g(x) dx is an inner product on D.
Ark9: Exercises for MAT2400 Fourier series The exercises o this sheet cover the sectios 4.9 to 4.13. They are iteded for the groups o Thursday, April 12 ad Friday, March 30 ad April 13. NB: No group o
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More informationSolutions to home assignments (sketches)
Matematiska Istitutioe Peter Kumli 26th May 2004 TMA401 Fuctioal Aalysis MAN670 Applied Fuctioal Aalysis 4th quarter 2003/2004 All documet cocerig the course ca be foud o the course home page: http://www.math.chalmers.se/math/grudutb/cth/tma401/
More informationSingular Continuous Measures by Michael Pejic 5/14/10
Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable
More informationSolutions to Tutorial 3 (Week 4)
The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial Week 4 MATH2962: Real ad Complex Aalysis Advaced Semester 1, 2017 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationMATH 312 Midterm I(Spring 2015)
MATH 3 Midterm I(Sprig 05) Istructor: Xiaowei Wag Feb 3rd, :30pm-3:50pm, 05 Problem (0 poits). Test for covergece:.. 3.. p, p 0. (coverges for p < ad diverges for p by ratio test.). ( coverges, sice (log
More informationMath 220A Fall 2007 Homework #2. Will Garner A
Math 0A Fall 007 Homewor # Will Garer Pg 3 #: Show that {cis : a o-egative iteger} is dese i T = {z œ : z = }. For which values of q is {cis(q): a o-egative iteger} dese i T? To show that {cis : a o-egative
More informationSUBSERIES CONVERGENCE AND SEQUENCE-EVALUATION CONVERGENCE. Min-Hyung Cho, Hong Taek Hwang and Won Sok Yoo. n t j x j ) = f(x 0 ) f(x j ) < +.
Kagweo-Kyugki Math. Jour. 6 (1998), No. 2, pp. 331 339 SUBSERIES CONVERGENCE AND SEQUENCE-EVALUATION CONVERGENCE Mi-Hyug Cho, Hog Taek Hwag ad Wo Sok Yoo Abstract. We show a series of improved subseries
More informationn=1 a n is the sequence (s n ) n 1 n=1 a n converges to s. We write a n = s, n=1 n=1 a n
Series. Defiitios ad first properties A series is a ifiite sum a + a + a +..., deoted i short by a. The sequece of partial sums of the series a is the sequece s ) defied by s = a k = a +... + a,. k= Defiitio
More informationDupuy Complex Analysis Spring 2016 Homework 02
Dupuy Complex Aalysis Sprig 206 Homework 02. (CUNY, Fall 2005) Let D be the closed uit disc. Let g be a sequece of aalytic fuctios covergig uiformly to f o D. (a) Show that g coverges. Solutio We have
More informationChapter 7 Isoperimetric problem
Chapter 7 Isoperimetric problem Recall that the isoperimetric problem (see the itroductio its coectio with ido s proble) is oe of the most classical problem of a shape optimizatio. It ca be formulated
More informationReal Variables II Homework Set #5
Real Variables II Homework Set #5 Name: Due Friday /0 by 4pm (at GOS-4) Istructios: () Attach this page to the frot of your homework assigmet you tur i (or write each problem before your solutio). () Please
More informationSTRONG QUASI-COMPLETE SPACES
Volume 1, 1976 Pages 243 251 http://topology.aubur.edu/tp/ STRONG QUASI-COMPLETE SPACES by Raymod F. Gittigs Topology Proceedigs Web: http://topology.aubur.edu/tp/ Mail: Topology Proceedigs Departmet of
More informationSPECTRUM OF THE DIRECT SUM OF OPERATORS
Electroic Joural of Differetial Equatios, Vol. 202 (202), No. 20, pp. 8. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.ut.edu ftp ejde.math.txstate.edu SPECTRUM OF THE DIRECT SUM
More information1 Convergence in Probability and the Weak Law of Large Numbers
36-752 Advaced Probability Overview Sprig 2018 8. Covergece Cocepts: i Probability, i L p ad Almost Surely Istructor: Alessadro Rialdo Associated readig: Sec 2.4, 2.5, ad 4.11 of Ash ad Doléas-Dade; Sec
More informationProposition 2.1. There are an infinite number of primes of the form p = 4n 1. Proof. Suppose there are only a finite number of such primes, say
Chater 2 Euclid s Theorem Theorem 2.. There are a ifiity of rimes. This is sometimes called Euclid s Secod Theorem, what we have called Euclid s Lemma beig kow as Euclid s First Theorem. Proof. Suose to
More informationA Proof of Birkhoff s Ergodic Theorem
A Proof of Birkhoff s Ergodic Theorem Joseph Hora September 2, 205 Itroductio I Fall 203, I was learig the basics of ergodic theory, ad I came across this theorem. Oe of my supervisors, Athoy Quas, showed
More informationb i u x i U a i j u x i u x j
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 9 N ov. 1 6, 2 0 0 9 ) S ecod- Order Elliptic Equatios: Weak S olutios 1. Defiitios. I this ad the followig two lectures we will study the boudary value problem Here
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationMATH 112: HOMEWORK 6 SOLUTIONS. Problem 1: Rudin, Chapter 3, Problem s k < s k < 2 + s k+1
MATH 2: HOMEWORK 6 SOLUTIONS CA PRO JIRADILOK Problem. If s = 2, ad Problem : Rudi, Chapter 3, Problem 3. s + = 2 + s ( =, 2, 3,... ), prove that {s } coverges, ad that s < 2 for =, 2, 3,.... Proof. The
More informationChapter 3 Inner Product Spaces. Hilbert Spaces
Chapter 3 Ier Product Spaces. Hilbert Spaces 3. Ier Product Spaces. Hilbert Spaces 3.- Defiitio. A ier product space is a vector space X with a ier product defied o X. A Hilbert space is a complete ier
More informationOn the behavior at infinity of an integrable function
O the behavior at ifiity of a itegrable fuctio Emmauel Lesige To cite this versio: Emmauel Lesige. O the behavior at ifiity of a itegrable fuctio. The America Mathematical Mothly, 200, 7 (2), pp.75-8.
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More informationLecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces
Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such
More informationHomework 4. x n x X = f(x n x) +
Homework 4 1. Let X ad Y be ormed spaces, T B(X, Y ) ad {x } a sequece i X. If x x weakly, show that T x T x weakly. Solutio: We eed to show that g(t x) g(t x) g Y. It suffices to do this whe g Y = 1.
More informationMath 116 Practice for Exam 3
Math 6 Practice for Eam 3 Geerated April 4, 26 Name: SOLUTIONS Istructor: Sectio Number:. This eam has questios. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur
More informationOn Topologically Finite Spaces
saqartvelos mecierebata erovuli aademiis moambe, t 9, #, 05 BULLETIN OF THE GEORGIAN NATIONAL ACADEMY OF SCIENCES, vol 9, o, 05 Mathematics O Topologically Fiite Spaces Giorgi Vardosaidze St Adrew the
More informationReal Numbers R ) - LUB(B) may or may not belong to B. (Ex; B= { y: y = 1 x, - Note that A B LUB( A) LUB( B)
Real Numbers The least upper boud - Let B be ay subset of R B is bouded above if there is a k R such that x k for all x B - A real umber, k R is a uique least upper boud of B, ie k = LUB(B), if () k is
More informationChapter 2. Periodic points of toral. automorphisms. 2.1 General introduction
Chapter 2 Periodic poits of toral automorphisms 2.1 Geeral itroductio The automorphisms of the two-dimesioal torus are rich mathematical objects possessig iterestig geometric, algebraic, topological ad
More informationON THE FUZZY METRIC SPACES
The Joural of Mathematics ad Computer Sciece Available olie at http://www.tjmcs.com The Joural of Mathematics ad Computer Sciece Vol.2 No.3 2) 475-482 ON THE FUZZY METRIC SPACES Received: July 2, Revised:
More informationA Characterization of Compact Operators by Orthogonality
Australia Joural of Basic ad Applied Scieces, 5(6): 253-257, 211 ISSN 1991-8178 A Characterizatio of Compact Operators by Orthogoality Abdorreza Paahi, Mohamad Reza Farmai ad Azam Noorafa Zaai Departmet
More informationMa 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5
Ma 42: Itroductio to Lebesgue Itegratio Solutios to Homework Assigmet 5 Prof. Wickerhauser Due Thursday, April th, 23 Please retur your solutios to the istructor by the ed of class o the due date. You
More informationMath 508 Exam 2 Jerry L. Kazdan December 9, :00 10:20
Math 58 Eam 2 Jerry L. Kazda December 9, 24 9: :2 Directios This eam has three parts. Part A has 8 True/False questio (2 poits each so total 6 poits), Part B has 5 shorter problems (6 poits each, so 3
More information1 Introduction. 1.1 Notation and Terminology
1 Itroductio You have already leared some cocepts of calculus such as limit of a sequece, limit, cotiuity, derivative, ad itegral of a fuctio etc. Real Aalysis studies them more rigorously usig a laguage
More informationPAPER : IIT-JAM 2010
MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure
More information