Problem Set 7: Solutions Math 201A: Fall 2016
|
|
- Linette McLaughlin
- 7 years ago
- Views:
Transcription
1 Problem Set 7: Solutions Math 21A: Fall 216 Problem 1. (a) A subset A X of a metric space X is nowhere dense in X if Ā = i.e., the interior of the closure is empty. Show that A is nowhere dense if and only if Āc is an open dense subset of X. (b) Which of the following sets are nowhere dense in R: A = {1/n : n N}; B = Q (, 1); C = Cantor set? (c) Show that the Baire category theorem implies that a complete metric space is not a countable union of closed, nowhere dense sets. (d) Show that a Hamel basis of an infinite-dimensional Banach space is uncountable. (a) The set Āc is open since Ā is closed. For any set B X, we have B c = [ ] c {F : F B c closed} = {G : G B open} = (B ) c. Hence, setting B = Ā, we get that (Āc ) = (Ā ) c, so Āc is dense in X if and only if Ā =. (b) The set A is nowhere dense since Ā = {} A has no interior points. The set B is not nowhere dense since B = [, 1] and B = (, 1). The Cantor set C is nowhere dense, since C = C and C contains no intervals so C =. (c) This follows directly from de Morgan s law. If F α is closed and nowhere dense, then G α = Fα c is open and dense, so G α and Fα = ( G α ) c X. (d) Suppose for contradiction that a Banach space X has a countably infinite Hamel basis {e n : n N}. Let E n = e 1,..., e n denote the linear span of the first n basis vectors. Then E n = X. However, E n is closed, since it is a finite-dimensional subspace, and E n is nowhere dense, since a finite-dimensional space is not a neighborhood of any point in X. The Baire category theorem implies that E n X, which is a contradiction. 1
2 Problem 2. Let X = M N where M, N are closed subspaces of a Banach space X with M N = {} and M + N = X. Define the projection P : X X of X onto M along N by P x = y where x = y + z with y M and z N. Prove that P is bounded. Let (x n ) be a sequence in X such that x n x and P x n y. If x n = y n + z n where y n M and z n N, then y n + z n x and y n y, so z n x y. Since M is closed, y M, and since N is closed, z = x y N. It follows that x = y + z where y M and z N, so P x = y, meaning that P is closed. The closed graph theorem implies that P is bounded. 2
3 Problem 3. (a) Let K : X X be a bounded linear operator on a Banach space X with K < 1. Show that I K has a bounded inverse given by the uniformly convergent series (I K) 1 = I + K + K 2 + K Also show that (I K) 1 1/(1 K ). (b) Suppose that k : [, 1] [, 1] R is a continuous function with sup { k(x, y) : (x, y) [, 1] [, 1]} < 1. Use (a) to show that the Fredholm integral equation of the second kind u(x) k(x, y)u(y) dy = f(x) x 1, has a unique solution u C([, 1]) for every f C([, 1]). Express the solution u as a series, and write out explicitly an approximation of u(x) valid for small k(x, y), up to terms of cubic order in k. (c) Show that the result in (b) can also be obtained from a contraction mapping iteration u n+1 = T u n with u = f where T : C([, 1]) C([, 1]) is defined by (T u)(x) = k(x, y)u(y) dy + f(x). (a) The series K n converges absolutely, since K n K n 1 1 K, n= n= so it converges in B(X) since B(X) is a Banach space. For every N N, we have ( N ) ( N ) (I K) K n = K n (I K) = I K N+1. n= n= If x X, then K N+1 x as N, and ( ) ( ) (I K) K n x = K n (I K)x = x, so n= Kn = (I K) 1, and (1 K) 1 1/(1 K ). 3
4 (b) Define K : C([, 1]) C([, 1]) by Then K < 1 since Ku(x) = k(x, y)f(y) dy. Ku θ u, θ = sup k(x, y) < 1, x,y [,1] so by (a) the equation has a unique solution u = (I K) 1 f. The expansion u = f + Kf + K 2 f + K 3 f +... gives u(x) = f(x) + k 2 (x, y) = k 3 (x, y) = + k(x, y)f(y) dy + k(x, s)k(s, y) ds, k 3 (x, y)f(y) dy +..., k(x, s)k(s, t)k(t, y) dsdt. k 2 (x, y)f(y) dy (c) The map T u = f + Ku is a contraction on C([, 1]), since T u T v = K(u v) θ u v, so the contraction mapping theorem implies that there is a unique solution of u = f + Ku. It follows by induction that the contraction mapping iterates u N+1 = T u N, starting with with u = f, are given by u N = N n= Kn f. Remark. The series for (I K) 1 is called the Neumann series. It shows that the invertible operators form an open set in B(X). The Neumann series is also used in problems from wave propagation and quantum mechanics (where it is called the Born approximation) to describe the scattering of waves by nonuniformities or a potential. Successive terms in the series describe contributions to the solution from singly, doubly, triply,... scattered waves. 4
5 Problem 4. Define the multiplication operator Φ : C([, 1]) C([, 1]) associated with a function φ C([, 1]) by Φf = φ f. (a) Equip C([, 1]) with the sup-norm f = sup x [,1] f(x). Show that Φ = φ. If (Φ n ) is a sequence of multiplication operators, show that Φ n strongly in B (C([, 1]), ) if and only if Φ n uniformly. (b) Equip C([, 1]) with the one-norm f 1 = f(x) dx. Give an example of a sequence of functions (φ n ) in C([, 1]) with associated multi- plication operators (Φ n ) such that Φ n strongly but not uniformly in B (C([, 1]), 1 ). (a) We have so Φ φ. Φf = sup φ(x)f(x) φ f, x [,1] Let 1 C([, 1]) denote the function 1(x) = 1. Then Φ Φ1 1 = φ. Thus, Φ = φ. Furthermore, Φ = Φ1. If Φ n strongly, then Φ n 1, which implies that Φ n, so Φ n uniformly. (b) Choose ψ C([, )) such that ψ 1 and { 1 if x 1, ψ(x) = if 2 x <. Define φ n C([, 1]) by φ n (x) = ψ(nx). Then φ n (x) dx 2 n, φ 2 n(x) dx 1 n. 5
6 If f C([, 1]), then there exists M > such that f(x) M for all x [, 1]. Hence Φ n f 1 = so Φ n strongly as n. On the other hand, Φ n 1 Φ nφ n 1 φ n 1 = so Φ n uniformly as n. φ n (x) f(x) dx 2M n, φ2 n(x) dx φ n(x) dx 1 2, 6
7 Problem 5. Let K : X X be a compact linear operator on an infinitedimensional Banach space X. If K is one-to-one, prove that the range of K is not closed. Suppose that K : X X is a compact operator on a Banach space X and M = ran K is closed. Then M is a Banach space and K : X M is one-to-one and onto. The open mapping theorem implies that K is open, so K is a homeomorphism between X and M. Let B X be the closed unit ball in X. Then K : B K(B) is a homeomorphism, and K(B) is compact since K is compact and K(B) X is closed. It follows that B is compact, which implies that X is finite-dimensional. 7
8 Problem 6. Let 1 < p <. Define the Hölder conjugate p of p by 1 p + 1 p = 1. (a) Prove Young s inequality: If a, b, then ab ap p + bp p. (b) Let l p = {(x n ) : x n R and (x n ) p < }, where ( ) 1/p (x n ) p = x n p. Prove Hölder s inequality: If x = (x n ) l p and y = (y n ) l p, then x n y n x p y p. (c) Show that (l p ) = l p. (a) Young s inequality is trivial if ab =, so assume that a, b >. By Lemma 1, the function log x is concave in x >, since (log x) = 1/x 2 <. It follows that for a, b >, log a + log b = 1 p log ap + 1 ( 1 p log bp log p ap + 1 ) p bp. Taking the exponential of this inequality gives the result. (b) First, consider finite sequences x = (x 1,..., x n ), y = (y 1,..., y n ) in R n, and assume that x p = ( n x k p ) 1/p, y p = ( n y k p ) 1/p are nonzero, otherwise the result is trivial. For any α, β >, x k y k α β x k p pα p + y k p p β p, 8
9 so n x k y k β pα p 1 n x k p + α p β p 1 n y k p. Choosing α = x p, β = y p, we get Hölder s inequality: n n x k y k x k y k 1 p x p y p + 1 p x p y p = x p y p. The inequality for x l p, y l p n. then follows by taking the limit as (c) For y l p, define a linear functional F : l p R by F (x) = x n y n. Hölder s inequality implies that F is bounded and F y p. Conversely, define x = (x n ) by x n = (sgn y n )y p 1 n. Since ( p 1 = p 1 1 ) = p p p, we have x l p with x p = y p /p p, so F F (x) x p = which proves that F = y p. Suppose that F (l p ). Let y p y p /p p = y p, y n = F (e n ) where e n = (,...,, 1,... ) is the nth basis vector in l p. If x = (y p 1 1,..., y p 1 n,... ), x p = 9 ( n ) 1/p y k p,
10 Then x belongs to the subspace c of sequences that have finitely many nonzero terms, and by linearity n F (x) = y k p. It follows that ( n ) 1/p y k p = which shows that y l p. F (x) x p F, Since c is dense in l p and F is bounded, F : c R extends by continuity to a unique linear functional F : l p R given by F (x) = x n y n, which shows that (l p ) = l p. The same argument shows that (l 1 ) = l. The proof fails for (l ) because c is not dense in l, and l 1 is strictly contained in (l ). Lemma 1. If f : (a, b) R is twice differentiable and f, then f is convex. Proof. First, suppose that f(y) f(x) + f (x)(y x) for all x, y (a, b), meaning that the graph of f lies above every tangent line. Let x, y (a, b) and z = tx + (1 t)y for t 1. Then f(x) f(z) + f (z)(x z), and it follows that f(y) f(z) + f (z)(y z), tf(x) + (1 t)f(y) f(z) + f (z) [t(x z) + (1 t)(y z)] = f(z), so f is convex. Now suppose that f. If x, y (a, b), then by Taylor s theorem with Lagrange remainder, there exists c between x, y such that f(y) = f(x) + f (x)(y x) f (c)(y x) 2 f(x) + f (x)(y x), so the graph of f lies above its tangent lines, and f is convex. 1
SOLUTIONS 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 information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
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 informationBANACH 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 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 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 information9 More on differentiation
Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......
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 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 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 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 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 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 informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
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 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 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 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 informationHow To Prove The 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 informationSOLUTIONS TO ASSIGNMENT 1 MATH 576
SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
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 informationCritical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.
Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =
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 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 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 informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
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 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 informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
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 informationReference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.
5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2
More information1 Local Brouwer degree
1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
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. 2005-03-14 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
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 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 informationSample Induction Proofs
Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given
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 informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,
More informationTHE CONTRACTION MAPPING THEOREM
THE CONTRACTION MAPPING THEOREM KEITH CONRAD 1. Introduction Let f : X X be a mapping from a set X to itself. We call a point x X a fixed point of f if f(x) = x. For example, if [a, b] is a closed interval
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 non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationPoint Set Topology. A. Topological Spaces and Continuous Maps
Point Set Topology A. Topological Spaces and Continuous Maps Definition 1.1 A topology on a set X is a collection T of subsets of X satisfying the following axioms: T 1.,X T. T2. {O α α I} T = α IO α T.
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 005-06-15 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationDifferential Operators and their Adjoint Operators
Differential Operators and their Adjoint Operators Differential Operators inear functions from E n to E m may be described, once bases have been selected in both spaces ordinarily one uses the standard
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
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 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 informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
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 informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
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 Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
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 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 informationIntroduction to Topology
Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................
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 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 informationLectures 5-6: Taylor Series
Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,
More informationThe sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].
Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
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 informationLecture Notes on Measure Theory and Functional Analysis
Lecture Notes on Measure Theory and Functional Analysis P. Cannarsa & T. D Aprile Dipartimento di Matematica Università di Roma Tor Vergata cannarsa@mat.uniroma2.it daprile@mat.uniroma2.it aa 2006/07 Contents
More informationCHAPTER 1 BASIC TOPOLOGY
CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is
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 non-empty
More informationMetric Spaces. Lecture Notes and Exercises, Fall 2015. M.van den Berg
Metric Spaces Lecture Notes and Exercises, Fall 2015 M.van den Berg School of Mathematics University of Bristol BS8 1TW Bristol, UK mamvdb@bristol.ac.uk 1 Definition of a metric space. Let X be a set,
More informationZORN S LEMMA AND SOME APPLICATIONS
ZORN S LEMMA AND SOME APPLICATIONS KEITH CONRAD 1. Introduction Zorn s lemma is a result in set theory that appears in proofs of some non-constructive existence theorems throughout mathematics. We will
More informationINTRODUCTION TO TOPOLOGY
INTRODUCTION TO TOPOLOGY ALEX KÜRONYA In preparation January 24, 2010 Contents 1. Basic concepts 1 2. Constructing topologies 13 2.1. Subspace topology 13 2.2. Local properties 18 2.3. Product topology
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 informationConsumer Theory. The consumer s problem
Consumer Theory The consumer s problem 1 The Marginal Rate of Substitution (MRS) We define the MRS(x,y) as the absolute value of the slope of the line tangent to the indifference curve at point point (x,y).
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 Stat-Math Unit, Indian Statistical Institute, 203 B.T. Road, Calcutta 700035,
More informationAP CALCULUS AB 2009 SCORING GUIDELINES
AP CALCULUS AB 2009 SCORING GUIDELINES Question 5 x 2 5 8 f ( x ) 1 4 2 6 Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected points in
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 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. SÁNCHEZ-PÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationGROUPS ACTING ON A SET
GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationLEARNING OBJECTIVES FOR THIS CHAPTER
CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional
More informationDescriptive Set Theory
Descriptive Set Theory David Marker Fall 2002 Contents I Classical Descriptive Set Theory 2 1 Polish Spaces 2 2 Borel Sets 14 3 Effective Descriptive Set Theory: The Arithmetic Hierarchy 27 4 Analytic
More information(Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties
Lecture 1 Convex Sets (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry
More information5. Linear algebra I: dimension
5. Linear algebra I: dimension 5.1 Some simple results 5.2 Bases and dimension 5.3 Homomorphisms and dimension 1. Some simple results Several observations should be made. Once stated explicitly, the proofs
More informationMath 104: Introduction to Analysis
Math 104: Introduction to Analysis Evan Chen UC Berkeley Notes for the course MATH 104, instructed by Charles Pugh. 1 1 August 29, 2013 Hard: #22 in Chapter 1. Consider a pile of sand principle. You wish
More informationIncreasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.
1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationWalrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.
Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
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 641-643, UAB 1999-2000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σ-algebras. A σ-algebra in X is a non-empty collection of subsets
More informationVector Spaces 4.4 Spanning and Independence
Vector Spaces 4.4 and Independence October 18 Goals Discuss two important basic concepts: Define linear combination of vectors. Define Span(S) of a set S of vectors. Define linear Independence of a set
More informationMathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More informationLebesgue Measure on R n
8 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 information17.3.1 Follow the Perturbed Leader
CS787: Advanced Algorithms Topic: Online Learning Presenters: David He, Chris Hopman 17.3.1 Follow the Perturbed Leader 17.3.1.1 Prediction Problem Recall the prediction problem that we discussed in class.
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More information