TERENCE TAO S AN EPSILON OF ROOM CHAPTER 3 EXERCISES. 1. Exercise 1.3.1
|
|
- Jeremy McDonald
- 5 years ago
- Views:
Transcription
1 TRNC TAO S AN PSILON OF ROOM CHAPTR 3 RCISS KLLR VANDBOGRT 1. xercise We merely consider the inclusion f f, viewed as an element of L p (, χ, µ), where all nonmeasurable subnull sets are given measure 0. Linearity is trivial. Surjectivity is also immediate, since the preimage is merely the function f itself. It remains to prove injectivity, so assume f 0. Then, Supp(f) A for some null set A. Integrating over A, we find that f = 0 a.e, which proves injectivity. 2. xercise (i). Note that whenever p < 1 and x 1, x x p. Then, 1 = f f + g + g f + g ( f ) p ( g + f + g f + g = f p + g p ( f + g ) p ) p = ( f + g ) p f p + g p Combining this with the triangle inequality and integrating, f + g p p f p p + g p p Date: September 17,
2 2 KLLR VANDBOGRT (ii). Note first that x x p is concave for p < 1. By homogeneity, it is of no loss of generality to assume that f p + g p = 1, so that for F p = G p = 1, f = (1 θ)f, g = θg, θ (0, 1) So that by concavity, (1 θ)f + θg p (1 θ) F p + θ G p Integrating yields f + g p 1 = f p + g p As asserted. (iii). We may again assume f p + g p = 1, so that for F p = G p = 1, By part (i), f = (1 θ)f, g = θg, θ (0, 1) f + g p ( (1 θ) p + θ p) 1/p Since x x 1/p is convex, ( (1 θ) p + θ p) 1/p (1 θ) (1 θ) 1 1/p + θ θ 1 1/p = (1 θ) 2 1/p + θ 2 1/p Optimizing in θ, we find that the minimum is achieved for θ = 1/2, implying ( 1 ) 2 1/p ( 1 2 1/p f + g p + 2 2) = 2 1/p 1 Which yields our optimal constant as 2 1/p 1, as asserted.
3 TRNC TAO S AN PSILON OF ROOM CHAPTR 3 RCISS 3 (iv). Note that by strict convexity/concavity, equality holds if and only if g = cf, c R. In the p = 1 case, we merely require that f and g always have the same sign (that is, fg 0). 3. xercise Suppose first that is a norm. If B denotes our unit ball, let x, y B, t (0, 1): (1 t)x + ty (1 t) x + t y 1 t + t = 1 So that B is convex. Conversely, we merely use contraposition. Then there exist points x and y B such that the triangle inequality does not hold. By homogeneity, we may assume that x = (1 t)x, y = ty for x, y B and that x + y = 1. If we consider the line segment through x and y, we see that for θ = t, (1 t)x + ty = x + y > x + y = 1 So that B is not convex, whence the result. 4. xercise [ Define A 0 := [1, ], A n := Supp( f ), we see: Yielding σ-finiteness. Supp(f) = 1 n+1, 1 n ]. Noting that Supp(f) = f 1 (A n ) n=0 5. xercise If f = 0, f 0 trivially. Assume now that f 0. For sufficiently small ɛ > 0, consider S ɛ := {x f(x) f ɛ}
4 4 KLLR VANDBOGRT By the previous problem, we may assume without loss of generality that µ(s ɛ ) <. Then, Taking the limit inferior, ( f p ( f ɛ) p dµ S ɛ = ( f ɛ)µ(s ɛ ) 1/p lim inf p f p f ɛ ) 1/p As ɛ > 0, is arbitrary, lim inf p f. Now, as f L p 0 L, Hölder s inequality yields for all p > p 0 : Letting p, p p 0 p f p f f p 0p p 0 lim sup f p f p Combining with the reverse inequality, we deduce lim p f p = f As asserted. Suppose now that f / L. Then, for every n > 0, there exists a set n and ɛ > 0 such that µ( n ) ɛ > 0 and f(x) > n for every x T ɛ. We see: Letting p, nɛ 1/p = nµ( n ) 1/p ( ) 1/p < f p dµ f p n n < lim inf p f p for all integers n. Hence, letting n, lim p f p =.
5 TRNC TAO S AN PSILON OF ROOM CHAPTR 3 RCISS 5 Define d(f, g) := f g. 6. xercise Homogeneity: d(cf, cg) = cf cg = c f g = c d(f, g) Triangle Inequality: d(f, h) = f h f g + g h = d(f, g) + d(g, h) Separation: d(f, g) = 0 f g = 0 f = g Symmetry: d(f, g) = f g = g f = d(g, f) Translation Invariance: d(f + h, g + h) = (f + h) (g + h) = f g = d(f, g) Conversely, suppose we have a translation invariant homogeneous metric d : V V [0, ]. Define f := d(f, 0). This choice is clearly unique, since any definition with respect to a nonzero basepoint loses homogeneity. It remains only to show the triangle inequality: f + g = d(f + g, 0) = d(f, g) d(f, 0) + d(0, g) = d(f, 0) + d(g, 0) = f + g So that defines a unique norm.
6 6 KLLR VANDBOGRT 7. xercise Assume first that V is complete. Given an absolutely convergent series n=1 f n, the sequence of partial sums s N is Cauchy. But f n f n n=1 So that N n=1 f n is bounded by s N. As the s N are Cauchy, we deduce that N n=1 f n is Cauchy. By completeness, this sequence converges, so that n=1 f n exists. Conversely, suppose that any absolutely convergent sum converges conditionally. Let f n be a Cauchy sequence, and extract a subsequence f nk such that n=1 f nk+1 f nk < 1 2 k for k N. Then, obviously k=1 f n k+1 f nk converges. By assumption, this implies k=1 f n k+1 f nk converges as well. But this sum in telescoping with limit lim k f nk f n1, so we deduce that f nk f for some f. It remains to show that f n f, but as f n is Cauchy: f n f f n f nk + f nk f 0 as n, k, so that f n f, implying that every Cauchy sequence is convergent, so V is complete. 8. xercise If f L, f is essentially bounded. Hence there exists a sequence of simple functions increasing to f (by the simple approximation theorem), and density is trivial. Note that the space of simple functions with finite measure is not dense in L. To see this, merely consider the characteristic function χ R. This has χ R = 1. Choosing a sequence
7 TRNC TAO S AN PSILON OF ROOM CHAPTR 3 RCISS 7 of simple functions s n with finite support, we see that for every n, s n vanishes outside of some sufficiently large set, so that χ R s n = 1 for all n. Therefore, this set cannot possibly be dense. 9. xercise numerate the generators of our σ algebra Ω by B = { 1, 2,... }. One immediately sees that { finite χ n n B} is dense in the space of characteristic function. Since Q is also dense in the reals, we see that S := span Q {χ n n B} is dense in the space of simple functions with finite measure support, denoted S 0, which in turn is dense in L p. But this implies S = S 0 and S 0 = L p From which we immediately see that S is dense in L p as well. But S is countable, so we deduce that L p is separable. For p =, consider the family C := {χ [ r,r] r R + } Then for any two distinct x, y C, x y = 1. But then L is certainly not separable, as no sequence of distinct elements could ever converge to an element of C.
8 8 KLLR VANDBOGRT 10. xercise Young s inequality is established by concavity, and the case of equality for strict concavity will occur precisely when a p = cb q. By homogeneity, we may assume f p = g q = c = 1. Then, Yielding the result. f(x)g(x) = 1 p f(x) p + 1 q g(x) q = fg 1 = 1 = f p g q Let f L p, q < p. Then ( f q 11. xercise ) 1/q 1/p ( ) 1/p 1dµ f p dµ = µ() 1/q 1/p f p So f L q. constant. By the previous problem, equality occurs when f is a 12. xercise Since q > p, we may apply the reverse H ólder inequality to find f p f q µ() 1/p 1/q By assumption, µ() m for every in our σ-algebra, so we rearrange the above inequality to find f q m 1/q 1/p f p So that f L q if f L p. quality holds again for f const.
9 TRNC TAO S AN PSILON OF ROOM CHAPTR 3 RCISS 9 We have: f(x) p dµ = ( 13. xercise f pθ f (1 θ)p d]mu = f pθ p 1 f p(1 θ) p 0 Taking pth roots in the above, whence the result. By Hölder s, ( ) ) f pθ p pθ ( 1 p1 pθ ( dµ ) ) f (1 θ)p p p(1 θ) 0 p (1 θ)p dµ 0 f p f θ p 1 f 1 θ p xercise so that f p p µ() 1 p/p 0 f p p 0 lim sup f p p µ() For the reverse inequality, lim inf f p dµ lim inf f p dµ lim inf f p dµ = dµ = µ() Hence we conclude that lim f p p = µ(). (Fatou s)
L p Spaces and Convexity
L p Spaces and Convexity These notes largely follow the treatments in Royden, Real Analysis, and Rudin, Real & Complex Analysis. 1. Convex functions Let I R be an interval. For I open, we say a function
More informationIt follows from the above inequalities that for c C 1
3 Spaces L p 1. Appearance of normed spaces. In this part we fix a measure space (, A, µ) (we may assume that µ is complete), and consider the A - measurable functions on it. 2. For f L 1 (, µ) set f 1
More informationMATH MEASURE THEORY AND FOURIER ANALYSIS. Contents
MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure
More informationReal Analysis Notes. Thomas Goller
Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................
More informationl(y j ) = 0 for all y j (1)
Problem 1. The closed linear span of a subset {y j } of a normed vector space is defined as the intersection of all closed subspaces containing all y j and thus the smallest such subspace. 1 Show that
More informationIt follows from the above inequalities that for c C 1
3 Spaces L p 1. In this part we fix a measure space (, A, µ) (we may assume that µ is complete), and consider the A -measurable functions on it. 2. For f L 1 (, µ) set f 1 = f L 1 = f L 1 (,µ) = f dµ.
More informationABSTRACT INTEGRATION CHAPTER ONE
CHAPTER ONE ABSTRACT INTEGRATION Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Suggestions and errors are invited and can be mailed
More information2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is
More informationLEBESGUE MEASURE AND L2 SPACE. Contents 1. Measure Spaces 1 2. Lebesgue Integration 2 3. L 2 Space 4 Acknowledgments 9 References 9
LBSGU MASUR AND L2 SPAC. ANNI WANG Abstract. This paper begins with an introduction to measure spaces and the Lebesgue theory of measure and integration. Several important theorems regarding the Lebesgue
More informationThree hours THE UNIVERSITY OF MANCHESTER. 24th January
Three hours MATH41011 THE UNIVERSITY OF MANCHESTER FOURIER ANALYSIS AND LEBESGUE INTEGRATION 24th January 2013 9.45 12.45 Answer ALL SIX questions in Section A (25 marks in total). Answer THREE of the
More informationFunctional Analysis, Stein-Shakarchi Chapter 1
Functional Analysis, Stein-Shakarchi Chapter 1 L p spaces and Banach Spaces Yung-Hsiang Huang 018.05.1 Abstract Many problems are cited to my solution files for Folland [4] and Rudin [6] post here. 1 Exercises
More information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More informationTHEOREMS, ETC., FOR MATH 515
THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every
More informationd(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N
Problem 1. Let f : A R R have the property that for every x A, there exists ɛ > 0 such that f(t) > ɛ if t (x ɛ, x + ɛ) A. If the set A is compact, prove there exists c > 0 such that f(x) > c for all x
More information(1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define
Homework, Real Analysis I, Fall, 2010. (1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define ρ(f, g) = 1 0 f(x) g(x) dx. Show that
More informationExamples of Dual Spaces from Measure Theory
Chapter 9 Examples of Dual Spaces from Measure Theory We have seen that L (, A, µ) is a Banach space for any measure space (, A, µ). We will extend that concept in the following section to identify an
More information1. Let A R be a nonempty set that is bounded from above, and let a be the least upper bound of A. Show that there exists a sequence {a n } n N
Applied Analysis prelim July 15, 216, with solutions Solve 4 of the problems 1-5 and 2 of the problems 6-8. We will only grade the first 4 problems attempted from1-5 and the first 2 attempted from problems
More informationMATH 426, TOPOLOGY. p 1.
MATH 426, TOPOLOGY THE p-norms In this document we assume an extended real line, where is an element greater than all real numbers; the interval notation [1, ] will be used to mean [1, ) { }. 1. THE p
More informationLebesgue Integration: A non-rigorous introduction. What is wrong with Riemann integration?
Lebesgue Integration: A non-rigorous introduction What is wrong with Riemann integration? xample. Let f(x) = { 0 for x Q 1 for x / Q. The upper integral is 1, while the lower integral is 0. Yet, the function
More informationMeasure Theory on Topological Spaces. Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond
Measure Theory on Topological Spaces Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond May 22, 2011 Contents 1 Introduction 2 1.1 The Riemann Integral........................................ 2 1.2 Measurable..............................................
More informationExercise Solutions to Functional Analysis
Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n
More informationIntroduction to Real Analysis Alternative Chapter 1
Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces
More informationChapter 5. Measurable Functions
Chapter 5. Measurable Functions 1. Measurable Functions Let X be a nonempty set, and let S be a σ-algebra of subsets of X. Then (X, S) is a measurable space. A subset E of X is said to be measurable if
More informationMATHS 730 FC Lecture Notes March 5, Introduction
1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists
More informationProblem Set 5: Solutions Math 201A: Fall 2016
Problem Set 5: s Math 21A: Fall 216 Problem 1. Define f : [1, ) [1, ) by f(x) = x + 1/x. Show that f(x) f(y) < x y for all x, y [1, ) with x y, but f has no fixed point. Why doesn t this example contradict
More informationNormed Vector Spaces and Double Duals
Normed Vector Spaces and Double Duals Mathematics 481/525 In this note we look at a number of infinite-dimensional R-vector spaces that arise in analysis, and we consider their dual and double dual spaces
More information212a1214Daniell s integration theory.
212a1214 Daniell s integration theory. October 30, 2014 Daniell s idea was to take the axiomatic properties of the integral as the starting point and develop integration for broader and broader classes
More informationMath 5051 Measure Theory and Functional Analysis I Homework Assignment 3
Math 551 Measure Theory and Functional Analysis I Homework Assignment 3 Prof. Wickerhauser Due Monday, October 12th, 215 Please do Exercises 3*, 4, 5, 6, 8*, 11*, 17, 2, 21, 22, 27*. Exercises marked with
More informationREAL AND COMPLEX ANALYSIS
REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any
More informationPROBLEMS. (b) (Polarization Identity) Show that in any inner product space
1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization
More informationAn introduction to some aspects of functional analysis
An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms
More informationCHAPTER 1. Metric Spaces. 1. Definition and examples
CHAPTER Metric Spaces. Definition and examples Metric spaces generalize and clarify the notion of distance in the real line. The definitions will provide us with a useful tool for more general applications
More informationNotes on Distributions
Notes on Distributions Functional Analysis 1 Locally Convex Spaces Definition 1. A vector space (over R or C) is said to be a topological vector space (TVS) if it is a Hausdorff topological space and the
More information18.175: Lecture 3 Integration
18.175: Lecture 3 Scott Sheffield MIT Outline Outline Recall definitions Probability space is triple (Ω, F, P) where Ω is sample space, F is set of events (the σ-algebra) and P : F [0, 1] is the probability
More informationReview of measure theory
209: Honors nalysis in R n Review of measure theory 1 Outer measure, measure, measurable sets Definition 1 Let X be a set. nonempty family R of subsets of X is a ring if, B R B R and, B R B R hold. bove,
More informationOverview of normed linear spaces
20 Chapter 2 Overview of normed linear spaces Starting from this chapter, we begin examining linear spaces with at least one extra structure (topology or geometry). We assume linearity; this is a natural
More informationRiesz Representation Theorems
Chapter 6 Riesz Representation Theorems 6.1 Dual Spaces Definition 6.1.1. Let V and W be vector spaces over R. We let L(V, W ) = {T : V W T is linear}. The space L(V, R) is denoted by V and elements of
More informationMATH 140B - HW 5 SOLUTIONS
MATH 140B - HW 5 SOLUTIONS Problem 1 (WR Ch 7 #8). If I (x) = { 0 (x 0), 1 (x > 0), if {x n } is a sequence of distinct points of (a,b), and if c n converges, prove that the series f (x) = c n I (x x n
More information1 Inner Product Space
Ch - Hilbert Space 1 4 Hilbert Space 1 Inner Product Space Let E be a complex vector space, a mapping (, ) : E E C is called an inner product on E if i) (x, x) 0 x E and (x, x) = 0 if and only if x = 0;
More informationTHE STONE-WEIERSTRASS THEOREM AND ITS APPLICATIONS TO L 2 SPACES
THE STONE-WEIERSTRASS THEOREM AND ITS APPLICATIONS TO L 2 SPACES PHILIP GADDY Abstract. Throughout the course of this paper, we will first prove the Stone- Weierstrass Theroem, after providing some initial
More informationAppendix A Functional Analysis
Appendix A Functional Analysis A.1 Metric Spaces, Banach Spaces, and Hilbert Spaces Definition A.1. Metric space. Let X be a set. A map d : X X R is called metric on X if for all x,y,z X it is i) d(x,y)
More information6. Duals of L p spaces
6 Duals of L p spaces This section deals with the problem if identifying the duals of L p spaces, p [1, ) There are essentially two cases of this problem: (i) p = 1; (ii) 1 < p < The major difference between
More informationMEASURE AND INTEGRATION: LECTURE 15. f p X. < }. Observe that f p
L saes. Let 0 < < and let f : funtion. We define the L norm to be ( ) / f = f dµ, and the sae L to be C be a measurable L (µ) = {f : C f is measurable and f < }. Observe that f = 0 if and only if f = 0
More informationProbability and Measure
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 84 Paper 4, Section II 26J Let (X, A) be a measurable space. Let T : X X be a measurable map, and µ a probability
More informationMeasurable functions are approximately nice, even if look terrible.
Tel Aviv University, 2015 Functions of real variables 74 7 Approximation 7a A terrible integrable function........... 74 7b Approximation of sets................ 76 7c Approximation of functions............
More informationAnalysis Comprehensive Exam Questions Fall 2008
Analysis Comprehensive xam Questions Fall 28. (a) Let R be measurable with finite Lebesgue measure. Suppose that {f n } n N is a bounded sequence in L 2 () and there exists a function f such that f n (x)
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More informationPartial Solutions to Folland s Real Analysis: Part I
Partial Solutions to Folland s Real Analysis: Part I (Assigned Problems from MAT1000: Real Analysis I) Jonathan Mostovoy - 1002142665 University of Toronto January 20, 2018 Contents 1 Chapter 1 3 1.1 Folland
More informationBased on the Appendix to B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge University press,
NOTE ON ABSTRACT RIEMANN INTEGRAL Based on the Appendix to B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge University press, 2003. a. Definitions. 1. Metric spaces DEFINITION 1.1. If
More informationCombinatorics in Banach space theory Lecture 12
Combinatorics in Banach space theory Lecture The next lemma considerably strengthens the assertion of Lemma.6(b). Lemma.9. For every Banach space X and any n N, either all the numbers n b n (X), c n (X)
More informationFunctional Analysis MATH and MATH M6202
Functional Analysis MATH 36202 and MATH M6202 1 Inner Product Spaces and Normed Spaces Inner Product Spaces Functional analysis involves studying vector spaces where we additionally have the notion of
More informationProblem set 4, Real Analysis I, Spring, 2015.
Problem set 4, Real Analysis I, Spring, 215. (18) Let f be a measurable finite-valued function on [, 1], and suppose f(x) f(y) is integrable on [, 1] [, 1]. Show that f is integrable on [, 1]. [Hint: Show
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week 2 November 6 November Deadline to hand in the homeworks: your exercise class on week 9 November 13 November Exercises (1) Let X be the following space of piecewise
More information1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),
Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer
More informationCONTENTS. 4 Hausdorff Measure Introduction The Cantor Set Rectifiable Curves Cantor Set-Like Objects...
Contents 1 Functional Analysis 1 1.1 Hilbert Spaces................................... 1 1.1.1 Spectral Theorem............................. 4 1.2 Normed Vector Spaces.............................. 7 1.2.1
More informationSummary of Real Analysis by Royden
Summary of Real Analysis by Royden Dan Hathaway May 2010 This document is a summary of the theorems and definitions and theorems from Part 1 of the book Real Analysis by Royden. In some areas, such as
More information2) Let X be a compact space. Prove that the space C(X) of continuous real-valued functions is a complete metric space.
University of Bergen General Functional Analysis Problems with solutions 6 ) Prove that is unique in any normed space. Solution of ) Let us suppose that there are 2 zeros and 2. Then = + 2 = 2 + = 2. 2)
More informationIntegral Jensen inequality
Integral Jensen inequality Let us consider a convex set R d, and a convex function f : (, + ]. For any x,..., x n and λ,..., λ n with n λ i =, we have () f( n λ ix i ) n λ if(x i ). For a R d, let δ a
More informationCompact operators on Banach spaces
Compact operators on Banach spaces Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto November 12, 2017 1 Introduction In this note I prove several things about compact
More informationReal Variables: Solutions to Homework 9
Real Variables: Solutions to Homework 9 Theodore D Drivas November, 20 xercise 0 Chapter 8, # : For complex-valued, measurable f, f = f + if 2 with f i real-valued and measurable, we have f = f + i f 2
More informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
More informationLECTURE NOTES FOR , FALL 2004
LECTURE NOTES FOR 18.155, FALL 2004 RICHARD B. MELROSE Contents Introduction 1 1. Continuous functions 2 2. Measures and σ algebras 10 3. Measureability of functions 16 4. Integration 19 5. Hilbert space
More informationP-adic Functions - Part 1
P-adic Functions - Part 1 Nicolae Ciocan 22.11.2011 1 Locally constant functions Motivation: Another big difference between p-adic analysis and real analysis is the existence of nontrivial locally constant
More informationGeneralized Orlicz spaces and Wasserstein distances for convex concave scale functions
Bull. Sci. math. 135 (2011 795 802 www.elsevier.com/locate/bulsci Generalized Orlicz spaces and Wasserstein distances for convex concave scale functions Karl-Theodor Sturm Institut für Angewandte Mathematik,
More informationB. Appendix B. Topological vector spaces
B.1 B. Appendix B. Topological vector spaces B.1. Fréchet spaces. In this appendix we go through the definition of Fréchet spaces and their inductive limits, such as they are used for definitions of function
More informationREAL ANALYSIS I HOMEWORK 4
REAL ANALYSIS I HOMEWORK 4 CİHAN BAHRAN The questions are from Stein and Shakarchi s text, Chapter 2.. Given a collection of sets E, E 2,..., E n, construct another collection E, E 2,..., E N, with N =
More informationMA677 Assignment #3 Morgan Schreffler Due 09/19/12 Exercise 1 Using Hölder s inequality, prove Minkowski s inequality for f, g L p (R d ), p 1:
Exercise 1 Using Hölder s inequality, prove Minkowski s inequality for f, g L p (R d ), p 1: f + g p f p + g p. Proof. If f, g L p (R d ), then since f(x) + g(x) max {f(x), g(x)}, we have f(x) + g(x) p
More informationCHAPTER I THE RIESZ REPRESENTATION THEOREM
CHAPTER I THE RIESZ REPRESENTATION THEOREM We begin our study by identifying certain special kinds of linear functionals on certain special vector spaces of functions. We describe these linear functionals
More informationIntroduction and Preliminaries
Chapter 1 Introduction and Preliminaries This chapter serves two purposes. The first purpose is to prepare the readers for the more systematic development in later chapters of methods of real analysis
More informationLECTURE NOTES FOR , FALL Contents. Introduction. 1. Continuous functions
LECTURE NOTES FOR 18.155, FALL 2002 RICHARD B. MELROSE Contents Introduction 1 1. Continuous functions 1 2. Measures and σ-algebras 9 3. Integration 16 4. Hilbert space 27 5. Test functions 30 6. Tempered
More informationSolutions: Problem Set 4 Math 201B, Winter 2007
Solutions: Problem Set 4 Math 2B, Winter 27 Problem. (a Define f : by { x /2 if < x
More informationLocally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem
56 Chapter 7 Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem Recall that C(X) is not a normed linear space when X is not compact. On the other hand we could use semi
More informationLebesgue Integration on R n
Lebesgue Integration on R n The treatment here is based loosely on that of Jones, Lebesgue Integration on Euclidean Space We give an overview from the perspective of a user of the theory Riemann integration
More information08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms
(February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops
More informationLecture 4 Lebesgue spaces and inequalities
Lecture 4: Lebesgue spaces and inequalities 1 of 10 Course: Theory of Probability I Term: Fall 2013 Instructor: Gordan Zitkovic Lecture 4 Lebesgue spaces and inequalities Lebesgue spaces We have seen how
More informationMath 118B Solutions. Charles Martin. March 6, d i (x i, y i ) + d i (y i, z i ) = d(x, y) + d(y, z). i=1
Math 8B Solutions Charles Martin March 6, Homework Problems. Let (X i, d i ), i n, be finitely many metric spaces. Construct a metric on the product space X = X X n. Proof. Denote points in X as x = (x,
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationconverges as well if x < 1. 1 x n x n 1 1 = 2 a nx n
Solve the following 6 problems. 1. Prove that if series n=1 a nx n converges for all x such that x < 1, then the series n=1 a n xn 1 x converges as well if x < 1. n For x < 1, x n 0 as n, so there exists
More informationMaths 212: Homework Solutions
Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then
More informationSet, functions and Euclidean space. Seungjin Han
Set, functions and Euclidean space Seungjin Han September, 2018 1 Some Basics LOGIC A is necessary for B : If B holds, then A holds. B A A B is the contraposition of B A. A is sufficient for B: If A holds,
More informationNotions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy
Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.
More informationMeasure and Integration: Solutions of CW2
Measure and Integration: s of CW2 Fall 206 [G. Holzegel] December 9, 206 Problem of Sheet 5 a) Left (f n ) and (g n ) be sequences of integrable functions with f n (x) f (x) and g n (x) g (x) for almost
More informationIntroduction to Functional Analysis
Introduction to Functional Analysis Carnegie Mellon University, 21-640, Spring 2014 Acknowledgements These notes are based on the lecture course given by Irene Fonseca but may differ from the exact lecture
More informationNumerical Sequences and Series
Numerical Sequences and Series Written by Men-Gen Tsai email: b89902089@ntu.edu.tw. Prove that the convergence of {s n } implies convergence of { s n }. Is the converse true? Solution: Since {s n } is
More informationMEASURE THEORY AND LEBESGUE INTEGRAL 15
MASUR THORY AND LBSGU INTGRAL 15 Proof. Let 2Mbe such that µ() = 0, and f 1 (x) apple f 2 (x) apple and f n (x) =f(x) for x 2 c.settingg n = c f n and g = c f, we observe that g n = f n a.e. and g = f
More informationDefinition 6.1. A metric space (X, d) is complete if every Cauchy sequence tends to a limit in X.
Chapter 6 Completeness Lecture 18 Recall from Definition 2.22 that a Cauchy sequence in (X, d) is a sequence whose terms get closer and closer together, without any limit being specified. In the Euclidean
More information(U) =, if 0 U, 1 U, (U) = X, if 0 U, and 1 U. (U) = E, if 0 U, but 1 U. (U) = X \ E if 0 U, but 1 U. n=1 A n, then A M.
1. Abstract Integration The main reference for this section is Rudin s Real and Complex Analysis. The purpose of developing an abstract theory of integration is to emphasize the difference between the
More informationProblem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationMath 140A - Fall Final Exam
Math 140A - Fall 2014 - Final Exam Problem 1. Let {a n } n 1 be an increasing sequence of real numbers. (i) If {a n } has a bounded subsequence, show that {a n } is itself bounded. (ii) If {a n } has a
More informationIntegration on Measure Spaces
Chapter 3 Integration on Measure Spaces In this chapter we introduce the general notion of a measure on a space X, define the class of measurable functions, and define the integral, first on a class of
More informationReal Analysis Problems
Real Analysis Problems Cristian E. Gutiérrez September 14, 29 1 1 CONTINUITY 1 Continuity Problem 1.1 Let r n be the sequence of rational numbers and Prove that f(x) = 1. f is continuous on the irrationals.
More informationThe Arzelà-Ascoli Theorem
John Nachbar Washington University March 27, 2016 The Arzelà-Ascoli Theorem The Arzelà-Ascoli Theorem gives sufficient conditions for compactness in certain function spaces. Among other things, it helps
More informationRecall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm
Chapter 13 Radon Measures Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm (13.1) f = sup x X f(x). We want to identify
More informationconsists of two disjoint copies of X n, each scaled down by 1,
Homework 4 Solutions, Real Analysis I, Fall, 200. (4) Let be a topological space and M be a σ-algebra on which contains all Borel sets. Let m, µ be two positive measures on M. Assume there is a constant
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationIntroduction to Hausdorff Measure and Dimension
Introduction to Hausdorff Measure and Dimension Dynamics Learning Seminar, Liverpool) Poj Lertchoosakul 28 September 2012 1 Definition of Hausdorff Measure and Dimension Let X, d) be a metric space, let
More informationProblem Set 1: Solutions Math 201A: Fall Problem 1. Let (X, d) be a metric space. (a) Prove the reverse triangle inequality: for every x, y, z X
Problem Set 1: s Math 201A: Fall 2016 Problem 1. Let (X, d) be a metric space. (a) Prove the reverse triangle inequality: for every x, y, z X d(x, y) d(x, z) d(z, y). (b) Prove that if x n x and y n y
More informationFUNCTIONAL ANALYSIS CHRISTIAN REMLING
FUNCTIONAL ANALYSIS CHRISTIAN REMLING Contents 1. Metric and topological spaces 2 2. Banach spaces 12 3. Consequences of Baire s Theorem 30 4. Dual spaces and weak topologies 34 5. Hilbert spaces 50 6.
More informationReal Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi
Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.
More informationTopological properties
CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological
More information