Midterm Exam Solutions February 27, 2009
|
|
|
- Branden McKenzie
- 5 years ago
- Views:
Transcription
1 Midterm Exam Solutions February 27, 2009 (24. Deine each o the ollowing statements. You may assume that everyone knows the deinition o a topological space and a linear space. (4 a. X is a compact topological space. Solution. Every open cover o X has a inite subcover. (4 b. X is a locally compact topological space. Solution. Every point in X has a compact neighborhood. (4 c. is a norm on a real linear space X. Solution. : X [ 0. satisies. x + y x + y, or all x, y X, 2. λx = λ x, or all x X, λ, 3. x = 0 x= 0. (4 d. C( X, F is an equicontinuous amily, where X is a topological space. Solution. For every x X and ε > 0, there exists a neighborhood U o x such that ( x ( y < ε or all y U and F. (4 e. X is the dual space o the complex Banach space X. Solution. X is the space o all bounded linear unctionals on X. (4. X is a meager topological space. Solution. X is the countable union o nowhere dense sets. Page o 5
2 (20 2. (0 a. Deine the product topology and state Tychono s Theorem. X α α is a amily o topological spaces, the product topology on X = X α is α A the weak topology generated by the projection maps π : X X. Solution. I { } A α α Alternatively, the product topology is the weakest topology such that the projection maps are continuous. Alternatively, the product topology is the weakest topology generated by the sets π, where α A and U is open in X α. α ( U Tychono s Theorem: I { X α } α A X = is compact in the product topology. X α α A is any amily o compact topological spaces, then (0 b. State the Baire Category Theorem, and give an example o a residual set which is not open. Solution. The Baire Category Theorem. Let X be a complete metric space. U is a sequence o open dense subsets o X, then = U n is dense in X. a. I { n} n b. X is not a countable union o nowhere dense sets. n= Example. Let X = with the usual topology, and let E = \. Then is meager, so E c is residual. E is not open, since E is dense. Page 2 o 5
3 (6 3. (8 a. State the Open Mapping Theorem or Banach spaces. Solution. I X and Y are Banach spaces and i T is a continuous linear surjection rom X to Y, then T is an open mapping. (8 b. Let X and Y be Banach spaces, and assume that T : X Y is a continuous linear bijection. Show that T is an isomorphism. Solution. I T is a continuous linear bijection, then T is a continuous linear surjection, so the Open Mapping Theorem implies that T is open. Since T is a bijection, T : Y X exists. Since T is open, the inverse image under T o every open set is open, so is continuous, hence T is an isomorphism. T Page 3 o 5
4 (20 4. Let be a seminorm on the linear space X, and let M { x X : x 0} subspace o X and that the map x + M x is a norm on X M. Solution. First we show that M is a subspace o X. Let x, y M and let a and b be scalars. Then ax + by ax + by = a x + b y = 0 so ax + by M, so M is a linear subspace o X. Next we show that the map x + M = x = =. Show that M is a is well-deined on X M. I x and y are in the same equivalence class, then x y M, so x y = 0. Thereore, x y x y = 0, so x = y, and hence x + M = x = y = y+ M Finally, we show that is a norm on X M. [ x+ M = x 0,, x + y+ M = x+ y x + y = x+ M + y+ M, ( x M x M x x x M λ + = λ + = λ = λ = λ +, x + M = 0 x = 0 x M x+ M = 0 in X M. Page 4 o 5
5 (20 5. Recall that, i ( X, ρ and (, continuous i there exists a contant λ 0 such that Y σ are metric spaces, a unction : X Y is called Lipschitz ( ( ( λρ( σ x, y x, y or all x, y X. Let X be a compact metric space, and let Y =, Show that the Lipschitz continuous unctions are dense in the space o continuous unctions with the uniorm norm. Solution. Let Lip ( X, denote the set o Lipschitz continuous real-valued unctions rom X to. First we show that the Lip ( X, is a linear subspace o C( X,. Since Lipschitz continuity implies continuity, we have that Lip ( X, C( X, ( X, g Lip,, and let ab,. Let Lipschitz constant or g. Then. To show that it is a linear subspace, let λ be the Lipschitz constant or, and let λ g be the ( a + bg ( x ( a + bg ( y = a( ( x ( y + b( g ( x g ( y a ( x ( y + b g( x g( y ( ( ( ( aλρ xy, + bλρ xy, = aλ + bλ ρ xy,, g g so ag + bg is Lipschitz continuous with Lipschitz constant a λ b λg +. Thereore, Lip ( X, is a linear subspace o C( X,. Next we show that Lip ( X, is a subalgebra o C( X,. Let, g Lip ( X, and λ g be as beore. Since X is compact, and g are bounded, so let ( x M g( x Mg, or all x X. Then ( ( = ( ( ( ( + ( ( ( ( ( x g( x g( y + g( y ( x ( y g x g y x g x x g y x g y y g y ( ( ( ( M λρ xy, + Mλρ xy, = Mλ + Mλ ρ xy,, g g g g so g is Lipschitz continuous with Lipschitz constant M λg + Mgλ., and let λ and Constant unctions are Lipschitz continuous with Lipschitz constant 0, so it remains only to Lip X, separates points. Let p and q be distinct points in X, and let show that ( Then ( p = 0 and ( q 0 that ( = ρ (, x x p, so separates p and q. Since the triangle inequality implies ( ( = ρ(, ρ(, ρ(, x y x p y p x y it ollows that is Lipschitz continuous with Lipschitz constant. The Stone-Weierstrass Theorem now implies that Lip ( X, is dense in (, C X. Page 5 o 5
Math General Topology Fall 2012 Homework 11 Solutions
Math 535 - General Topology Fall 2012 Homework 11 Solutions Problem 1. Let X be a topological space. a. Show that the following properties of a subset A X are equivalent. 1. The closure of A in X has empty
Math 209B Homework 2
Math 29B Homework 2 Edward Burkard Note: All vector spaces are over the field F = R or C 4.6. Two Compactness Theorems. 4. Point Set Topology Exercise 6 The product of countably many sequentally compact
1.4 The Jacobian of a map
1.4 The Jacobian of a map Derivative of a differentiable map Let F : M n N m be a differentiable map between two C 1 manifolds. Given a point p M we define the derivative of F at p by df p df (p) : T p
Functional 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
Midterm 1. Every element of the set of functions is continuous
Econ 200 Mathematics for Economists Midterm Question.- Consider the set of functions F C(0, ) dened by { } F = f C(0, ) f(x) = ax b, a A R and b B R That is, F is a subset of the set of continuous functions
Analysis Comprehensive Exam Questions Fall F(x) = 1 x. f(t)dt. t 1 2. tf 2 (t)dt. and g(t, x) = 2 t. 2 t
Analysis Comprehensive Exam Questions Fall 2. Let f L 2 (, ) be given. (a) Prove that ( x 2 f(t) dt) 2 x x t f(t) 2 dt. (b) Given part (a), prove that F L 2 (, ) 2 f L 2 (, ), where F(x) = x (a) Using
Functional Analysis (H) Midterm USTC-2015F haha Your name: Solutions
Functional Analysis (H) Midterm USTC-215F haha Your name: Solutions 1.(2 ) You only need to anser 4 out of the 5 parts for this problem. Check the four problems you ant to be graded. Write don the definitions
THEOREMS, 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
NAME: Mathematics 205A, Fall 2008, Final Examination. Answer Key
NAME: Mathematics 205A, Fall 2008, Final Examination Answer Key 1 1. [25 points] Let X be a set with 2 or more elements. Show that there are topologies U and V on X such that the identity map J : (X, U)
Chapter 3: Baire category and open mapping theorems
MA3421 2016 17 Chapter 3: Baire category and open mapping theorems A number of the major results rely on completeness via the Baire category theorem. 3.1 The Baire category theorem 3.1.1 Definition. A
E.7 Alaoglu s Theorem
E.7 Alaoglu s Theorem 359 E.7 Alaoglu s Theorem If X is a normed space, then the closed unit ball in X or X is compact if and only if X is finite-dimensional (Problem A.25). Even so, Alaoglu s Theorem
Normed and Banach Spaces
(August 30, 2005) Normed and Banach Spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ We have seen that many interesting spaces of functions have natural structures of Banach spaces:
Notes for Functional Analysis
Notes for Functional Analysis Wang Zuoqin (typed by Xiyu Zhai) September 29, 2015 1 Lecture 09 1.1 Equicontinuity First let s recall the conception of equicontinuity for family of functions that we learned
Functional Analysis Exercise Class
Functional Analysis Exercise Class Wee November 30 Dec 4: Deadline to hand in the homewor: your exercise class on wee December 7 11 Exercises with solutions Recall that every normed space X can be isometrically
Contents Ordered Fields... 2 Ordered sets and fields... 2 Construction of the Reals 1: Dedekind Cuts... 2 Metric Spaces... 3
Analysis Math Notes Study Guide Real Analysis Contents Ordered Fields 2 Ordered sets and fields 2 Construction of the Reals 1: Dedekind Cuts 2 Metric Spaces 3 Metric Spaces 3 Definitions 4 Separability
MAA6617 COURSE NOTES SPRING 2014
MAA6617 COURSE NOTES SPRING 2014 19. Normed vector spaces Let X be a vector space over a field K (in this course we always have either K = R or K = C). Definition 19.1. A norm on X is a function : X K
Problem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
Spaces of continuous functions
Chapter 2 Spaces of continuous functions 2.8 Baire s Category Theorem Recall that a subset A of a metric space (X, d) is dense if for all x X there is a sequence from A converging to x. An equivalent definition
Locally 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
Normed 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
Reflexivity of Locally Convex Spaces over Local Fields
Reflexivity of Locally Convex Spaces over Local Fields Tomoki Mihara University of Tokyo & Keio University 1 0 Introduction For any Hilbert space H, the Hermit inner product induces an anti C- linear isometric
ANALYSIS WORKSHEET II: METRIC SPACES
ANALYSIS WORKSHEET II: METRIC SPACES Definition 1. A metric space (X, d) is a space X of objects (called points), together with a distance function or metric d : X X [0, ), which associates to each pair
The weak topology of locally convex spaces and the weak-* topology of their duals
The weak topology of locally convex spaces and the weak-* topology of their duals Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto April 3, 2014 1 Introduction These notes
Chapter 2 Metric Spaces
Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics
Math 5052 Measure Theory and Functional Analysis II Homework Assignment 7
Math 5052 Measure Theory and Functional Analysis II Homework Assignment 7 Prof. Wickerhauser Due Friday, February 5th, 2016 Please do Exercises 3, 6, 14, 16*, 17, 18, 21*, 23*, 24, 27*. Exercises marked
1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3
Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,
MAT 578 FUNCTIONAL ANALYSIS EXERCISES
MAT 578 FUNCTIONAL ANALYSIS EXERCISES JOHN QUIGG Exercise 1. Prove that if A is bounded in a topological vector space, then for every neighborhood V of 0 there exists c > 0 such that tv A for all t > c.
Functional Analysis. Martin Brokate. 1 Normed Spaces 2. 2 Hilbert Spaces The Principle of Uniform Boundedness 32
Functional Analysis Martin Brokate Contents 1 Normed Spaces 2 2 Hilbert Spaces 2 3 The Principle of Uniform Boundedness 32 4 Extension, Reflexivity, Separation 37 5 Compact subsets of C and L p 46 6 Weak
Lecture Course. Functional Analysis
MATHEMATISCHES INSTITUT PROF. DR. PETER MÜLLER Summer Term 2013 Lecture Course Functional Analysis Typesetting by Kilian Lieret and Marcel Schaub If you find mistakes, I would appreciate getting a short
Bounded and continuous functions on a locally compact Hausdorff space and dual spaces
Chapter 6 Bounded and continuous functions on a locally compact Hausdorff space and dual spaces Recall that the dual space of a normed linear space is a Banach space, and the dual space of L p is L q where
MATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES
MATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES PETE L. CLARK 4. Metric Spaces (no more lulz) Directions: This week, please solve any seven problems. Next week, please solve seven more. Starred parts of
B. 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
Continuous Functions on Metric Spaces
Continuous Functions on Metric Spaces Math 201A, Fall 2016 1 Continuous functions Definition 1. Let (X, d X ) and (Y, d Y ) be metric spaces. A function f : X Y is continuous at a X if for every ɛ > 0
Lectures on Analysis John Roe
Lectures on Analysis John Roe 2005 2009 1 Lecture 1 About Functional Analysis The key objects of study in functional analysis are various kinds of topological vector spaces. The simplest of these are the
Selçuk Demir WS 2017 Functional Analysis Homework Sheet
Selçuk Demir WS 2017 Functional Analysis Homework Sheet 1. Let M be a metric space. If A M is non-empty, we say that A is bounded iff diam(a) = sup{d(x, y) : x.y A} exists. Show that A is bounded iff there
Problem Set. Problems on Unordered Summation. Math 5323, Fall Februray 15, 2001 ANSWERS
Problem Set Problems on Unordered Summation Math 5323, Fall 2001 Februray 15, 2001 ANSWERS i 1 Unordered Sums o Real Terms In calculus and real analysis, one deines the convergence o an ininite series
MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION. Problem 1
MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION ELEMENTS OF SOLUTION Problem 1 1. Let X be a Hausdorff space and K 1, K 2 disjoint compact subsets of X. Prove that there exist disjoint open sets U 1 and
GELFAND S THEOREM. Christopher McMurdie. Advisor: Arlo Caine. Spring 2004
GELFAND S THEOREM Christopher McMurdie Advisor: Arlo Caine Super Advisor: Dr Doug Pickrell Spring 004 This paper is a proof of the first part of Gelfand s Theorem, which states the following: Every compact,
Completeness and quasi-completeness. 1. Products, limits, coproducts, colimits
(April 24, 2014) Completeness and quasi-completeness Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/fun/notes 2012-13/07d quasi-completeness.pdf]
Your first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
MAT 544 Problem Set 2 Solutions
MAT 544 Problem Set 2 Solutions Problems. Problem 1 A metric space is separable if it contains a dense subset which is finite or countably infinite. Prove that every totally bounded metric space X is separable.
9.3 Graphing Functions by Plotting Points, The Domain and Range of Functions
9. Graphing Functions by Plotting Points, The Domain and Range o Functions Now that we have a basic idea o what unctions are and how to deal with them, we would like to start talking about the graph o
7 Complete metric spaces and function spaces
7 Complete metric spaces and function spaces 7.1 Completeness Let (X, d) be a metric space. Definition 7.1. A sequence (x n ) n N in X is a Cauchy sequence if for any ɛ > 0, there is N N such that n, m
CLASS NOTES FOR APRIL 14, 2000
CLASS NOTES FOR APRIL 14, 2000 Announcement: Section 1.2, Questions 3,5 have been deferred from Assignment 1 to Assignment 2. Section 1.4, Question 5 has been dropped entirely. 1. Review of Wednesday class
Math 216A. A gluing construction of Proj(S)
Math 216A. A gluing construction o Proj(S) 1. Some basic deinitions Let S = n 0 S n be an N-graded ring (we ollows French terminology here, even though outside o France it is commonly accepted that N does
Problem 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
Chapter 1. Introduction
Chapter 1 Introduction Functional analysis can be seen as a natural extension of the real analysis to more general spaces. As an example we can think at the Heine - Borel theorem (closed and bounded is
The projectivity of C -algebras and the topology of their spectra
The projectivity of C -algebras and the topology of their spectra Zinaida Lykova Newcastle University, UK Waterloo 2011 Typeset by FoilTEX 1 The Lifting Problem Let A be a Banach algebra and let A-mod
Problem 1: Compactness (12 points, 2 points each)
Final exam Selected Solutions APPM 5440 Fall 2014 Applied Analysis Date: Tuesday, Dec. 15 2014, 10:30 AM to 1 PM You may assume all vector spaces are over the real field unless otherwise specified. Your
FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets
FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES CHRISTOPHER HEIL 1. Compact Sets Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E X be
Analysis Preliminary Exam Workshop: Hilbert Spaces
Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H
MATH 113 SPRING 2015
MATH 113 SPRING 2015 DIARY Effective syllabus I. Metric spaces - 6 Lectures and 2 problem sessions I.1. Definitions and examples I.2. Metric topology I.3. Complete spaces I.4. The Ascoli-Arzelà Theorem
Continuity of convex functions in normed spaces
Continuity of convex functions in normed spaces In this chapter, we consider continuity properties of real-valued convex functions defined on open convex sets in normed spaces. Recall that every infinitedimensional
SOME EXERCISES FOR THE MINICOURSE: BANACH REPRESENTATIONS OF DYNAMICAL SYSTEMS. Contents 1. Exercises 1 2. Some Solutions 4
SOME EXERCISES FOR THE MINICOURSE: BANACH REPRESENTATIONS OF DYNAMICAL SYSTEMS MICHAEL MEGRELISHVILI Contents 1. Exercises 1 2. Some Solutions 4 1. Exercises Exercise 1.1. Show that the one point compactification
GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS
GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS CHRIS HENDERSON Abstract. This paper will move through the basics o category theory, eventually deining natural transormations and adjunctions
1 Categories, Functors, and Natural Transformations. Discrete categories. A category is discrete when every arrow is an identity.
MacLane: Categories or Working Mathematician 1 Categories, Functors, and Natural Transormations 1.1 Axioms or Categories 1.2 Categories Discrete categories. A category is discrete when every arrow is an
Math 5210, Definitions and Theorems on Metric Spaces
Math 5210, Definitions and Theorems on Metric Spaces Let (X, d) be a metric space. We will use the following definitions (see Rudin, chap 2, particularly 2.18) 1. Let p X and r R, r > 0, The ball of radius
Real Analysis Chapter 4 Solutions Jonathan Conder
2. Let x, y X and suppose that x y. Then {x} c is open in the cofinite topology and contains y but not x. The cofinite topology on X is therefore T 1. Since X is infinite it contains two distinct points
CHAPTER 5. The Topology of R. 1. Open and Closed Sets
CHAPTER 5 The Topology of R 1. Open and Closed Sets DEFINITION 5.1. A set G Ω R is open if for every x 2 G there is an " > 0 such that (x ", x + ") Ω G. A set F Ω R is closed if F c is open. The idea is
Real Analysis April f (t) dt. Show that g(x) 0 as x.
eal Analysis April Part A. Let f L ( and let g(x x f (t dt. Show that g(x as x. Let {x n } be a sequence in such that x n as n, and define h n (t : χ [xn, f (t. Now, given ε >, there is some N such that
Compact 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
Real Analysis, 2nd Edition, G.B.Folland Elements of Functional Analysis
Real Analysis, 2nd Edition, G.B.Folland Chapter 5 Elements of Functional Analysis Yung-Hsiang Huang 5.1 Normed Vector Spaces 1. Note for any x, y X and a, b K, x+y x + y and by ax b y x + b a x. 2. It
Analysis III. Exam 1
Analysis III Math 414 Spring 27 Professor Ben Richert Exam 1 Solutions Problem 1 Let X be the set of all continuous real valued functions on [, 1], and let ρ : X X R be the function ρ(f, g) = sup f g (1)
2 Some more things about locally convex tvs
2 Some more things about locally convex tvs Definition 2.1 A subset A of a tvs is called s-bounded iff, for any open neighborhood U of 0, there is a k K with A ku. Theorem 2.2 Compact sets are s-bounded,
CHAPTER V DUAL SPACES
CHAPTER V DUAL SPACES DEFINITION Let (X, T ) be a (real) locally convex topological vector space. By the dual space X, or (X, T ), of X we mean the set of all continuous linear functionals on X. By the
VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS
VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Recall that or prevarieties, we had criteria or being a variety or or being complete in terms o existence and uniqueness o limits, where
Chapter 1: Banach Spaces
321 2008 09 Chapter 1: Banach Spaces The main motivation for functional analysis was probably the desire to understand solutions of differential equations. As with other contexts (such as linear algebra
Functional 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......................................
Topological vectorspaces
(July 25, 2011) Topological vectorspaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ Natural non-fréchet spaces Topological vector spaces Quotients and linear maps More topological
Real 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..............................
(c) For each α R \ {0}, the mapping x αx is a homeomorphism of X.
A short account of topological vector spaces Normed spaces, and especially Banach spaces, are basic ambient spaces in Infinite- Dimensional Analysis. However, there are situations in which it is necessary
Functional Analysis HW #1
Functional Analysis HW #1 Sangchul Lee October 9, 2015 1 Solutions Solution of #1.1. Suppose that X
Lecture Notes in Functional Analysis
Lecture Notes in Functional Analysis Please report any mistakes, misprints or comments to aulikowska@mimuw.edu.pl LECTURE 1 Banach spaces 1.1. Introducton to Banach Spaces Definition 1.1. Let X be a K
1 Relative degree and local normal forms
THE ZERO DYNAMICS OF A NONLINEAR SYSTEM 1 Relative degree and local normal orms The purpose o this Section is to show how single-input single-output nonlinear systems can be locally given, by means o a
Continuous functions that are nowhere differentiable
Continuous functions that are nowhere differentiable S. Kesavan The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai - 600113. e-mail: kesh @imsc.res.in Abstract It is shown that the existence
Analysis Finite and Infinite Sets The Real Numbers The Cantor Set
Analysis Finite and Infinite Sets Definition. An initial segment is {n N n n 0 }. Definition. A finite set can be put into one-to-one correspondence with an initial segment. The empty set is also considered
Functional Analysis, Math 7320 Lecture Notes from August taken by Yaofeng Su
Functional Analysis, Math 7320 Lecture Notes from August 30 2016 taken by Yaofeng Su 1 Essentials of Topology 1.1 Continuity Next we recall a stronger notion of continuity: 1.1.1 Definition. Let (X, d
Part III. 10 Topological Space Basics. Topological Spaces
Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.
Review Notes for the Basic Qualifying Exam
Review Notes for the Basic Qualifying Exam Topics: Analysis & Linear Algebra Fall 2016 Spring 2017 Created by Howard Purpose: This document is a compilation of notes generated to prepare for the Basic
VALUATIVE CRITERIA BRIAN OSSERMAN
VALUATIVE CRITERIA BRIAN OSSERMAN Intuitively, one can think o separatedness as (a relative version o) uniqueness o limits, and properness as (a relative version o) existence o (unique) limits. It is not
Aliprantis, Border: Infinite-dimensional Analysis A Hitchhiker s Guide
aliprantis.tex May 10, 2011 Aliprantis, Border: Infinite-dimensional Analysis A Hitchhiker s Guide Notes from [AB2]. 1 Odds and Ends 2 Topology 2.1 Topological spaces Example. (2.2) A semimetric = triangle
NORMED ALGEBRAS OF DIFFERENTIABLE FUNCTIONS ON COMPACT PLANE SETS
NORMED ALGEBRAS O DIERENTIABLE UNCTIONS ON COMPACT PLANE SETS H. G. DALES AND J.. EINSTEIN Abstract. We investigate the completeness and completions of the normed algebras (D (1) (X), ) for perfect, compact
Part A. Metric and Topological Spaces
Part A Metric and Topological Spaces 1 Lecture A1 Introduction to the Course This course is an introduction to the basic ideas of topology and metric space theory for first-year graduate students. Topology
CONTENTS. 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
EXPANSIVE ALGEBRAIC ACTIONS OF DISCRETE RESIDUALLY FINITE AMENABLE GROUPS AND THEIR ENTROPY
EXPANSIVE ALGEBRAIC ACTIONS OF DISCRETE RESIDUALLY FINITE AMENABLE GROUPS AND THEIR ENTROPY CHRISTOPHER DENINGER AND KLAUS SCHMIDT Abstract. We prove an entropy ormula or certain expansive actions o a
Math General Topology Fall 2012 Homework 8 Solutions
Math 535 - General Topology Fall 2012 Homework 8 Solutions Problem 1. (Willard Exercise 19B.1) Show that the one-point compactification of R n is homeomorphic to the n-dimensional sphere S n. Note that
SEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Last quarter, we introduced the closed diagonal condition or a prevariety to be a prevariety, and the universally closed condition or a variety to be complete.
Homework Assignment #5 Due Wednesday, March 3rd.
Homework Assignment #5 Due Wednesday, March 3rd. 1. In this problem, X will be a separable Banach space. Let B be the closed unit ball in X. We want to work out a solution to E 2.5.3 in the text. Work
SEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN The notions o separatedness and properness are the algebraic geometry analogues o the Hausdor condition and compactness in topology. For varieties over the
Finite-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 (, )
Algebraic Geometry I Lectures 22 and 23
Algebraic Geometry I Lectures 22 and 23 Amod Agashe December 4, 2008 1 Fibered Products (contd..) Recall the deinition o ibered products g!θ X S Y X Y π 2 π 1 S By the universal mapping property o ibered
2.2 Annihilators, complemented subspaces
34CHAPTER 2. WEAK TOPOLOGIES, REFLEXIVITY, ADJOINT OPERATORS 2.2 Annihilators, complemented subspaces Definition 2.2.1. [Annihilators, Pre-Annihilators] Assume X is a Banach space. Let M X and N X. We
F 1 =. Setting F 1 = F i0 we have that. j=1 F i j
Topology Exercise Sheet 5 Prof. Dr. Alessandro Sisto Due to 28 March Question 1: Let T be the following topology on the real line R: T ; for each finite set F R, we declare R F T. (a) Check that T is a
Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012
Instructions: Answer all of the problems. Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012 Definitions (2 points each) 1. State the definition of a metric space. A metric space (X, d) is set
MA677 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
Functional Analysis I
Functional Analysis I Course Notes by Stefan Richter Transcribed and Annotated by Gregory Zitelli Polar Decomposition Definition. An operator W B(H) is called a partial isometry if W x = X for all x (ker
FUNCTIONAL 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.
MAT 771 FUNCTIONAL ANALYSIS HOMEWORK 3. (1) Let V be the vector space of all bounded or unbounded sequences of complex numbers.
MAT 771 FUNCTIONAL ANALYSIS HOMEWORK 3 (1) Let V be the vector space of all bounded or unbounded sequences of complex numbers. (a) Define d : V V + {0} by d(x, y) = 1 ξ j η j 2 j 1 + ξ j η j. Show that
(convex combination!). Use convexity of f and multiply by the common denominator to get. Interchanging the role of x and y, we obtain that f is ( 2M ε
1. Continuity of convex functions in normed spaces In this chapter, we consider continuity properties of real-valued convex functions defined on open convex sets in normed spaces. Recall that every infinitedimensional
I teach myself... Hilbert spaces
I teach myself... Hilbert spaces by F.J.Sayas, for MATH 806 November 4, 2015 This document will be growing with the semester. Every in red is for you to justify. Even if we start with the basic definition