x pn x qn 0 as n. Every convergent sequence is Cauchy. Not every Cauchy sequence in a normed space E converges to a vector in


 Rodger Paul
 2 years ago
 Views:
Transcription
1 78 CHAPTER 3. BANACH SPACES 3.2 Banach Spaces Cauchy Sequence. A sequence of vectors (x n ) in a normed space is a Cauchy sequence if for everyε > 0 there exists M N such that for all n, m M, x m x n <ε. Theorem A sequence (x n ) is a Cauchy sequence if and only if for any two subsequences (x pn ) and (x qn ) x pn x qn 0 as n. Every convergent sequence is Cauchy. Not every Cauchy sequence in a normed space E converges to a vector in E. Example. Incompleteness. Lemma Let (x n ) be a Cauchy sequence of vectors in a normed space. Then the sequence ( x n ) or real numbers converges. 1. We have x m x n x m x n. Thus ( x n ) is Cauchy. Every Cauchy sequence is bounded. Exercise. Banach Space. A normed space E is complete (or Banach space) if every Cauchy sequence in E converges to an element in E. Examples.R n andc n (with any norm) are complete. Theorem The spaces L p (Ω) with p 1 and l p are complete. Theorem Completeness of l 2. The space of complex sequences with the norm 2 is complete. No proof. mathphyshass1.tex; September 25, 2013; 22:56; p. 76
2 3.2. BANACH SPACES 79 Theorem Completeness of C(Ω). The space of complexvalued continuous functions on a closed bounded setω R n with the norm is complete. No proof. f = sup f (x) x Ω Convergent and Absolutely Convergent Series. A series n=1 x n converges in a normed space E if the sequence of partial sums s n = n k=1 x k converges in E. If n=1 x n <, then the series converges absolutely. An absolutely convergent series does not need to converge. Theorem A normed space is complete if and only if every absolutely convergent series converges. in functional analysis books. Theorem A closed vector subspace of a Banach space is a Banach space. no proof. Completion of Normed Spaces Let (E, ) be a normed space. A normed space (Ẽ, 1 ) is a completion of (E, ) if there exists a linear injectionφ : E Ẽ, such that 1. for every x E 2.Φ(E) is dense in Ẽ, 3. Ẽ is complete. x = Φ(x) 1, The space Ẽ is defined as the set of equivalence classes of Cauchy sequences in E. Recall that if (x n ) is Cauchy then the sequence x n converges. mathphyshass1.tex; September 25, 2013; 22:56; p. 77
3 80 CHAPTER 3. BANACH SPACES We define an equivalence relation of Cauchy sequences as follows. Two Cauchy sequences (x n ) and (y n ) in E are equivalent, (x n ) (y n ), if lim x n y n = 0. The set of Cauchy sequences (y n ) equivalent to a Cauchy sequence (x n ) is the equivalence class of (x n ) [(x n )]={(y n ) E (y n ) (x n )} Obviously we can identify the space E with the set of constant Cauchy sequences. Then the set of all equivalent classes is Ẽ=E/ ={[(x n )] (x n ) is Cauchy sequence in E} The addition and multiplication by scalars in Ẽ are defined by [(x n )]+[(y n )]=[(x n + y n )], λ[(x n )]=[(λx n )] The norm in Ẽ is defined by the limit [(x n )] 1 = lim n x n, which exists for every Cauchy sequence. This definition is consistent since for any two equivalent Cauchy sequences (x n ) and (y n ) [(x n )] 1 = [(y n )] 1 Exercise. The linear injectionφ : E Ẽis defined by a constant sequence ThenΦis onetoone. Exercise. Φ(x)=[(x n )] such that x n = x, n N. mathphyshass1.tex; September 25, 2013; 22:56; p. 78
4 3.2. BANACH SPACES 81 Obviously, x E, x = Φ(x) 1. Claim:Φ(E) is dense in Ẽ. Since every element [(x n )] of Ẽ is the limit of a sequence (Φ(x n )). Claim: Ẽ is complete. Let ( x n ) be a Cauchy sequence in Ẽ. Then (x n ) in E such that Φ(x n ) x n 1 < 1 n. Claim: (x n ) is Cauchy sequence in E. x n x m = Φ(x n ) Φ(x m ) 1 Φ(x n ) x n 1 + x n x m 1 + x m Φ(x m ) 1 x n x m n + 1 m. Next, let x=[(x n )]. Claim: x n x 1 0. x n x 1 x n Φ(x n ) 1 + Φ(x n ) x 1 < 1 n + Φ(x n) x 1 0. Homeomorphism. Two topological spaces E 1 and E 2 are homeomorphic if there exists a bijectionψ : E 1 E 2 from E 1 onto E 2 such that bothψ andψ 1 are continuous. mathphyshass1.tex; September 25, 2013; 22:56; p. 79
5 82 CHAPTER 3. BANACH SPACES Isomorphism of Normed Spaces. Two normed spaces (E 1, 1 ) and (E 2, 2 ) are isomorphic if there exists a linear homeomorphismψ : E 1 E 2 from E 1 onto E 2. Theorem Any two completions of a normed space are isomorphic. Read elsewhere. mathphyshass1.tex; September 25, 2013; 22:56; p. 80
6 3.3. LINEAR MAPPINGS Linear Mappings Let L : E 1 E 2 be a mapping from a vector space E 1 into a vector space E 2. If x E 1, then L(x) is the image of the vector x. If A E 1 is a subset of E 1, then the set is the image of the set A. L(A)={y E 2 y=l(x) for some x A} If B E 2 is a subset of E 2, then the set is the inverse image of the set B. L 1 (B)={x E 1 L(x) B} A mapping L : D(L) E 2 may be defined on a proper subset (called the domain) D(L) E 1 of the vector space E 1. The image of the domain, L(D(L)), of a mapping L is the range of L. That is the range of L is R(L)={y E 2 y=l(x) for some x D(L)}. The null space N(L) (or the kernel Ker(L)) of a mapping L is the set of all vectors in the domain D(L) which are mapped to zero, that is N(L)={x D(L) L(x)=0}. The graphγ(l) of a mapping L is the set of ordered pairs (x, L(x)), that is Γ(L)={(x, y) E 1 E 2 x D(L) and y=l(x)}. Continuous Mappings. A mapping f : E 1 E 2 from a normed space E 1 into a normed space E 2 is continuous at x 0 E 1 if any sequence (x n ) in E 1 converging to x 0 is mapped to a sequence f (x n ) in E 2 that converges to f (x 0 ). mathphyshass1.tex; September 25, 2013; 22:56; p. 81
7 84 CHAPTER 3. BANACH SPACES That is f is continuous at x 0 if x n x 0 0 implies f (x n ) f (x 0 ) 0. A mapping f : E 1 E 2 is continuous if it is continuous at every x E 1. Theorem The norm : E R in a normed space E is a continuous mapping from E intor. If x n x 0, then x n x x n x 0 Theorem Let f : E 1 E 2 be a mapping from a normed space E 1 into a normed space E 2. The following conditions are equivalent: 1. f is continuous. 2. The inverse image of any open set of E 2 is open in E The inverse image of any closed set of E 2 is closed in E 1. Exercise. Linear Mappings. Let S E 1 be a subset of a vector space E 1. A mapping L : S E 2 is linear if x, y S and α,β Fsuch thatαx+βy S, L(αx+βy)=αL(x)+βL(y). Theorem If S is not a vector subspace of E 1, then there is a unique extension of L : S E 2 to a linear mapping L : span S E 2 from the vector subspace span S to E 2. The extension L is defined by linearity. Thus, one can always assume that the domain of a linear mapping is a vector space. Theorem The range, the null space and the graph of a linear mapping are vector spaces. Exercise. mathphyshass1.tex; September 25, 2013; 22:56; p. 82
8 3.3. LINEAR MAPPINGS 85 For any linear mapping L, L(0)=0. Thus, 0 N(L) and the null space N(L) is always nonempty. Theorem A linear mapping L : E 1 E 2 from a normed space E 1 into a normed space E 2 is continuous if and only if it is continuous at a point. 1. Assume L is continuous at x 0 E Let x E 1 and (x n ) x. 3. Then (x n x+ x 0 ) x Thus L(x n ) L(x) = L(x n x+ x 0 ) L(x 0 ) 0 Bounded Linear Mappings. A linear mapping L : E 1 E 2 from a normed space E 1 into a normed space E 2 is bounded if there is a real number K R such that for all x E 1, L(x) K x. Theorem A linear mapping L : E 1 E 2 from a normed space E 1 into a normed space E 2 is continuous if and only if it is bounded. 1. (I). Assume that L is bounded. 2. Claim: L is continuous at Indeed, x n 0 implies 4. Hence, L is continuous. 5. (II). Assume that L is continuous. L(x n ) K x n 0 6. By contradiction, assume that L is unbounded. 7. Then, there is a sequence (x n ) in E 1 such that L(x n ) > n x n mathphyshass1.tex; September 25, 2013; 22:56; p. 83
9 86 CHAPTER 3. BANACH SPACES 8. Let 9. Then y n = x n n x n, n N y n = 1 n, and L(y n) > Then y n 0 but L(y n ) Thus, L is not continuous at zero. Remark. For linear mappings, continuity and uniform continuity are equivalent. The set L(E 1, E 2 ) of all linear mappings from a vector space E 1 into a vector space E 2 is a vector space with the addition and multiplication by scalars defined by (L 1 + L 2 )(x)=l 1 (x)+ L 2 (x), and (αl)(x)=αl(x). The set B(E 1, E 2 ) of all bounded linear mappings from a normed space E 1 into a normed space E 2 is a vector subspace of the space L(E 1, E 2 ). Theorem The space B(E 1, E 2 ) of all bounded linear mappings L : E 1 E 2 from a normed space E 1 into a normed space E 2 is a normed space with norm defined by 1. Obviously, L 0. L(x) L = sup x E 1,x 0 x 2. L =0 if and only if L=0. 3. Claim: L satisfies triangle inequality. 4. Let L 1, L 2 B(E 1, E 2 ). = sup L(x). x E 1, x =1 mathphyshass1.tex; September 25, 2013; 22:56; p. 84
10 3.3. LINEAR MAPPINGS Then L 1 + L 2 = sup L 1 (x)+ L 2 (x) x =1 sup L 1 (x) +sup L 2 (x) x =1 x =1 = L 1 + L 2 For any bounded linear mapping L : E 1 E 2 L(x) L x, x E 1. L is the least real number K such that L(x) K x for all x E 1. The norm defined by L = sup x E1, x =1 norm. L(x) is called the operator Convergence with respect to the operator norm is called the uniform convergence of operators. Strong convergence. A sequence of bounded linear mappings L n B(E 1, E 2 ) converges strongly to L B(E 1, E 2 ) if for every x E 1 we have L n (x) L(x) 0 as n. Theorem Uniform convergence implies strong convergence. Follows from Converse is not true. L n (x) L(x) L n L x Theorem Let E 1 be a normed space and E 2 be a Banach space. Then B(E 1, E 2 ) is a Banach space. see functional analysis books. mathphyshass1.tex; September 25, 2013; 22:56; p. 85
11 88 CHAPTER 3. BANACH SPACES Theorem Let E 1 be a normed space and E 2 be a Banach space. Let S E 1 be a subspace of E 1 and L : S E 2 be a continuous linear mapping from S into E 2. Then: 1. L has a unique extension to a continuous linear mapping L : S E 2 defined on the closure of the domain of the mapping L. 2. If S is dense in E 1, then L has a unique extension to a continuous linear mapping L : E 1 E 2. see functional analysis. Theorem Let E 1 and E 2 be normed spaces, S E 1 be a subspace of E 1 and L : S E 2 be a continuous linear mapping from S into E 2. Then: 1. the null space N(L) is a closed subspace of E If the domain S is a closed subspace of E 1, then the graphγ(l) of L is a closed subspace of E 1 E 2. Exercise. A bounded linear mapping L : E F from a normed space E into the scalar fieldfis called a functional. The space B(E,F) of functionals is called the dual space and denoted by E or E. The dual space is always a Banach space. sincefis complete. mathphyshass1.tex; September 25, 2013; 22:56; p. 86
12 3.4. BANACH FIXED POINT THEOREM Banach Fixed Point Theorem Contraction Mapping. Let E be a normed space and A Ebe a subset of E. A mapping f : A E from A into E is a contraction mapping if there exists a real numberα, such that 0<α<1and x, y A f (x) f (y) α x y. A contraction mapping is continuous. Exercise. If x, y A f (x) f (y) x y, then it is not necessarily a contraction since the contraction constantαmay not exist. Theorem Banach Fixed Point Theorem. Let E be a Banach space and A E be a closed subset of E. Let f : A A be a contraction mapping from A into A. Then there exists a unique z A such that f (z) = z. Examples. Homework Exercises: DM[9,10,11,12,14,21,23,26,31,34,36,37,38,39,42,44] mathphyshass1.tex; September 25, 2013; 22:56; p. 87
BANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationChapter 5. Banach Spaces
9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting on
More informationFinite dimensional topological vector spaces
Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the
More informationThese axioms must hold for all vectors ū, v, and w in V and all scalars c and d.
DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More informationNotes on metric spaces
Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More information16.3 Fredholm Operators
Lectures 16 and 17 16.3 Fredholm Operators A nice way to think about compact operators is to show that set of compact operators is the closure of the set of finite rank operator in operator norm. In this
More informationLet H and J be as in the above lemma. The result of the lemma shows that the integral
Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;
More information1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 0050615 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More information1 Normed vector spaces and convexity
B4a Banach spaces lectured by Dmitry Belyaev based on notes by B. Kirchheim and CJK Batty 1 Normed vector spaces and convexity Mods analysis on the real line derives many of its basic statements (algebra
More informationTHE BANACH CONTRACTION PRINCIPLE. Contents
THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationINTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES
INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. We study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the
More informationMatrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for
More informationn k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...
6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationMetric Spaces Joseph Muscat 2003 (Last revised May 2009)
1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of
More informationSOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationVector Spaces II: Finite Dimensional Linear Algebra 1
John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationOpen and Closed Sets
Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationLinear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.
Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a subvector space of V[n,q]. If the subspace of V[n,q]
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
More informationON A GLOBALIZATION PROPERTY
APPLICATIONES MATHEMATICAE 22,1 (1993), pp. 69 73 S. ROLEWICZ (Warszawa) ON A GLOBALIZATION PROPERTY Abstract. Let (X, τ) be a topological space. Let Φ be a class of realvalued functions defined on X.
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More informationTensor product of vector spaces
Tensor product of vector spaces Construction Let V,W be vector spaces over K = R or C. Let F denote the vector space freely generated by the set V W and let N F denote the subspace spanned by the elements
More informationDEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS
DEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsionfree abelian groups, one of isomorphism
More informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationLinear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold:
Linear Algebra A vector space (over R) is an ordered quadruple (V, 0, α, µ) such that V is a set; 0 V ; and the following eight axioms hold: α : V V V and µ : R V V ; (i) α(α(u, v), w) = α(u, α(v, w)),
More informationRate of growth of Dfrequently hypercyclic functions
Rate of growth of Dfrequently hypercyclic functions A. Bonilla Departamento de Análisis Matemático Universidad de La Laguna Hypercyclic Definition A (linear and continuous) operator T in a topological
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More informationRi and. i=1. S i N. and. R R i
The subset R of R n is a closed rectangle if there are n nonempty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an
More informationON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction
ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity
More informationMA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam
MA 4 LINEAR ALGEBRA C, Solutions to Second Midterm Exam Prof. Nikola Popovic, November 9, 6, 9:3am  :5am Problem (5 points). Let the matrix A be given by 5 6 5 4 5 (a) Find the inverse A of A, if it exists.
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationMatrix Representations of Linear Transformations and Changes of Coordinates
Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under
More information{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...
44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it
More informationFunctional analysis and its applications
Department of Mathematics, London School of Economics Functional analysis and its applications Amol Sasane ii Introduction Functional analysis plays an important role in the applied sciences as well as
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationSystems with Persistent Memory: the Observation Inequality Problems and Solutions
Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +
More informationVectors. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) Vectors Spring /
Vectors Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) Vectors Spring 2012 1 / 18 Introduction  Definition Many quantities we use in the sciences such as mass, volume, distance, can be expressed
More information1 Eigenvalues and Eigenvectors
Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x
More informationand s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space
RAL ANALYSIS A survey of MA 641643, UAB 19992000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σalgebras. A σalgebra in X is a nonempty collection of subsets
More informationAbout the inverse football pool problem for 9 games 1
Seventh International Workshop on Optimal Codes and Related Topics September 61, 013, Albena, Bulgaria pp. 15133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute
More informationDynamics for the solutions of the waterhammer equations
Dynamics for the solutions of the waterhammer equations J. Alberto Conejero (IUMPAUniversitat Politècnica de València). Joint work with C. Lizama (Universidad Santiado de Chile) and F. Ródenas (IUMPAUniversitat
More informationLINEAR DIFFERENTIAL EQUATIONS IN DISTRIBUTIONS
LINEAR DIFFERENTIAL EQUATIONS IN DISTRIBUTIONS LESLIE D. GATES, JR.1 1. Introduction. The main result of this paper is an extension of the analysis of Laurent Schwartz [l ] concerning the primitive of
More informationChapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces
Chapter 1. Metric Spaces Metric Spaces MA222 David Preiss d.preiss@warwick.ac.uk Warwick University, Spring 2008/2009 Definitions. A metric on a set M is a function d : M M R such that for all x, y, z
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More informationMathematics for Econometrics, Fourth Edition
Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents
More informationA Problem With The Rational Numbers
Solvability of Equations Solvability of Equations 1. In fields, linear equations ax + b = 0 are solvable. Solvability of Equations 1. In fields, linear equations ax + b = 0 are solvable. 2. Quadratic equations
More informationOn Nicely Smooth Banach Spaces
E extracta mathematicae Vol. 16, Núm. 1, 27 45 (2001) On Nicely Smooth Banach Spaces Pradipta Bandyopadhyay, Sudeshna Basu StatMath Unit, Indian Statistical Institute, 203 B.T. Road, Calcutta 700035,
More informationHedging bounded claims with bounded outcomes
Hedging bounded claims with bounded outcomes Freddy Delbaen ETH Zürich, Department of Mathematics, CH892 Zurich, Switzerland Abstract. We consider a financial market with two or more separate components
More informationNonArbitrage and the Fundamental Theorem of Asset Pricing: Summary of Main Results
Proceedings of Symposia in Applied Mathematics Volume 00, 1997 NonArbitrage and the Fundamental Theorem of Asset Pricing: Summary of Main Results Freddy Delbaen and Walter Schachermayer Abstract. The
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a nonempty
More informationSec 4.1 Vector Spaces and Subspaces
Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationConvex analysis and profit/cost/support functions
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m
More informationLecture 3: Linear Programming Relaxations and Rounding
Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can
More information( ) which must be a vector
MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationPROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume L, Number 3, September 2005 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK Abstract.
More informationLow upper bound of ideals, coding into rich Π 0 1 classes
Low upper bound of ideals, coding into rich Π 0 1 classes Antonín Kučera the main part is a joint project with T. Slaman Charles University, Prague September 2007, Chicago The main result There is a low
More informationA CHARACTERIZATION OF C k (X) FOR X NORMAL AND REALCOMPACT
Bol. Soc. Mat. Mexicana (3) Vol. 12, 2006 A CHARACTERIZATION OF C k (X) FOR X NORMAL AND REALCOMPACT F. MONTALVO, A. A. PULGARÍN AND B. REQUEJO Abstract. We present some internal conditions on a locally
More informationAdvanced Microeconomics
Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68 Part A. Basic decision and preference theory 1 Decisions
More information0 <β 1 let u(x) u(y) kuk u := sup u(x) and [u] β := sup
456 BRUCE K. DRIVER 24. Hölder Spaces Notation 24.1. Let Ω be an open subset of R d,bc(ω) and BC( Ω) be the bounded continuous functions on Ω and Ω respectively. By identifying f BC( Ω) with f Ω BC(Ω),
More informationEMBEDDING COUNTABLE PARTIAL ORDERINGS IN THE DEGREES
EMBEDDING COUNTABLE PARTIAL ORDERINGS IN THE ENUMERATION DEGREES AND THE ωenumeration DEGREES MARIYA I. SOSKOVA AND IVAN N. SOSKOV 1. Introduction One of the most basic measures of the complexity of a
More information4. Expanding dynamical systems
4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,
More information2 Complex Functions and the CauchyRiemann Equations
2 Complex Functions and the CauchyRiemann Equations 2.1 Complex functions In onevariable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More informationSurface bundles over S 1, the Thurston norm, and the Whitehead link
Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3manifold can fiber over the circle. In
More information1. Let X and Y be normed spaces and let T B(X, Y ).
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: NVP, Frist. 20050314 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationSubspaces of R n LECTURE 7. 1. Subspaces
LECTURE 7 Subspaces of R n Subspaces Definition 7 A subset W of R n is said to be closed under vector addition if for all u, v W, u + v is also in W If rv is in W for all vectors v W and all scalars r
More informationMATH 110 Spring 2015 Homework 6 Solutions
MATH 110 Spring 2015 Homework 6 Solutions Section 2.6 2.6.4 Let α denote the standard basis for V = R 3. Let α = {e 1, e 2, e 3 } denote the dual basis of α for V. We would first like to show that β =
More informationLinear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)
MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationORIENTATIONS. Contents
ORIENTATIONS Contents 1. Generators for H n R n, R n p 1 1. Generators for H n R n, R n p We ended last time by constructing explicit generators for H n D n, S n 1 by using an explicit nsimplex which
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More information1 Fixed Point Iteration and Contraction Mapping Theorem
1 Fixed Point Iteration and Contraction Mapping Theorem Notation: For two sets A,B we write A B iff x A = x B. So A A is true. Some people use the notation instead. 1.1 Introduction Consider a function
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
More informationMonotone maps of R n are quasiconformal
Monotone maps of R n are quasiconformal K. Astala, T. Iwaniec and G. Martin For Neil Trudinger Abstract We give a new and completely elementary proof showing that a δ monotone mapping of R n, n is K quasiconformal
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationx a x 2 (1 + x 2 ) n.
Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More information