THE INVERSE FUNCTION THEOREM

Size: px
Start display at page:

Download "THE INVERSE FUNCTION THEOREM"

Transcription

1 THE INVERSE FUNCTION THEOREM W. PATRICK HOOPER The implicit function theorem is the following result: Theorem 1. Let f be a C 1 function from a neighborhood of a point a R n into R n. Suppose A = Df(a) is invertible. Then, there is a neighborhood U of a so that the restriction f U : U f(u) is invertible and D(f 1 U )( f(x) ) = Df(x) 1 for all x U. We will pursue a proof of this fact using the contraction mapping theorem. (I m not sure this is the quickest path to a proof, but we are following this approach to demonstrate the technique of this sort of proof.) 1. A family of contractions The approach is motivated by Newton s method for finding roots of a polynomial (or other C 1 function) by iteratively improving approximations. We begin with a discussion of the idea of the proof. Our function f is C 1 at a R n with derivative A. Let y be a point nearby b = f(a), and let x be a point near a. We treat x as a guess for the value of f 1 (y ), and will try to improve this guess. Because f is C 1, Df is nearly A on a small neighborhood about a. So an approximation for the function f near x (which is near a) is given by the affine map L(x) = f(x ) + A(x x ). Because A is invertible, we can invert the function L. We think of L 1 as an approximation to f 1 near f(x ), and we have L 1 (y) = x + A 1( y f(x ) ). So our iteratively improved guess for the value of f 1 (y ) is obtained by replacing x with x 1 = L 1 (y ). The notation above is not ideal, so we will introduce new notation. Let N be a neighborhood of a on which f is C 1. We will denote by φ y the map x L 1 (y ) with L 1 depending on x as above. Getting rid of the asterisk, we have: φ y : N R n ; x x + A 1( y f(x) ). We record some basic properties of this function in the following propositions. First Proposition. The map φ b fixes a. Moreover, we have f(x) = y if and only if x is fixed by φ y. Proof. The first statement is a special case of the second. Observe that φ y (x) = x if and only if = φ y (x) x = A 1( y f(x) ). Date: October, 14. 1

2 W. PATRICK HOOPER By multiplying through by the invertible matrix A, we see that this is equivalent to = y f(x). The following explains that φ b is a very strong contraction locally near a. Proposition 3. For any c >, there is an r = r(c) > so that the closed ball Br (a) of radius r centered at a is contained in the domain of f and whenever x, y B r (a) we have φ b (x) φ b (y) c x y. We will make use of the idea of the operator norm in the proof. If M is an n n matrix, its operator norm is Mx M = max. x R n x The expression inside the maximum is invariant under scaling, so by continuity and compactness of the unit sphere, the supremum is realized at some point on this sphere. It easily follows that the operator norm has the following properties: The identity has operator norm 1. The quantity M varies continuously in M. M with equality if and only M is the zero matrix. cm = c M if M is n n and c R. M 1 +M M 1 + M and M 1 M M 1 M for every pair of n n matrices. Proof of Proposition 3. Choose an r so that f is C 1 on B r (a). Choose x, y B r (a). Observe that (1) φ b (x) φ b (y) = x y A 1( f(x) f(y) ) = A 1( f(x) f(y) A(x y) ). Consider the n-component functions of f, i.e., f(x) = ( f 1 (x), f (x),..., f n (x) ). Let z t = tx + (1 t)y for x [, 1]. The fundamental theorem of calculus in each coordinate tells us that 1 [ ( ) ] f i (x) f i (y) = Df zt (x y) dt. i By considering our integrals to be defined coordinate wise, we can write: f(x) f(y) = 1 Df ( z t ) (x y) dt. Plugging this into equation 1 and simplifying, we see: ( φ b (x) φ b (y) = A 1 1 () Df( ) ) z t (x y) dt A(x y) ( = A 1 1 [ ( ) ] ) Df zt A (x y) dt. Now we will prove the proposition while making use of the operator norm. Fix some c >. Since Df(a) = A, we have Df(a) A =. Because f is C 1 and by continuity of the operator norm, we can choose an r so that whenever z B r (a), we have Df(z) A c A 1.

3 THE INVERSE FUNCTION THEOREM 3 Now take x, y B r (a) and recall that with our definition above we have z t B r (a) for every t [, 1]. Then by definition of the operator norm, [ Df ( ) ] c z t A (x y) x y. A 1 We are integrating the vectors [ Df ( ) ] z t A (x y) from to 1, which you should think of as an average. Since every vector has a length uniformly bounded as above, the average has the same bound: 1 [ ( ) ] Df zt A (x y) dt c x y. A 1 We obtain the right side of equation by multiplying by A 1, so by using the definition of the operator norm, we have: φ b (x) φ b (y) 1 [ ( ) ] A 1 Df zt A (x y) dt c x y. We will now improve the prior proposition so that it holds for y near b. Proposition 4. Suppose < c < 1, and define r = r(c) > as in Proposition 3. (1) For any y R n, whenever x 1, x B r (a), we have φ y (x 1 ) φ y (x ) c x 1 x. () There is an open ball V = V (c) about b so that for any y V, the image φ y ( Br (a) ) is contained in the open ball B r (a). Proof. To see statement (1), choose x 1, x B r (a). Observe that It follows using Proposition 3 that φ y (x 1 ) φ y (x ) = φ b (x 1 ) φ b (x ). φ y (x 1 ) φ y (x ) = φ b (x 1 ) φ b (x ) c x 1 x. Now we consider statement (). Observe that the function (x, y) φ y (x) is jointly continuous. In particular, if V is a compact neighborhood of y, the restricted function B r (a) V R n ; (x, y) φ y (x) is uniformly continuous. Therefore, there is a δ > so that for each y V and each x B r (a), y b < δ implies φ y (x) φ b (x) < (1 c)r. Now recall that because φ b contracts B r (a) by a factor of c and fixes a, we have x B r (a) implies φ b (x) a c x a cr. Then by the triangle inequality, if y V and y b < δ, for any x B r (a), we have φ y (x) a φ y (x) φ b (x) + φ b (x) a < (1 c)r + cr = r. So, the conclusion of statement () follows by choosing V = V (c) to be an open ball about b that is contained in V B δ (b).

4 4 W. PATRICK HOOPER For concreteness take c = 1 in the above proposition. Then there is an r = r(c ) and a neighborhood V = V (c ) so that for any y V, φ y sends B r (a) into its interior and contracts distances by a factor of at least 1. As a consequence of the contraction mapping theorem, we see that for each y V (with V as in Proposition 4), there is a unique fixed point of the restriction of φ y to B r (a), and this fixed point lies in the open ball B r (a). Let g : V B r (a) be the map which sends each y V to this fixed point of φ y. As a direct consequence of Proposition : Corollary 5. We have f g(y) = y for every y V. We now turn our attention to proving that g is continuous. The key point is that the proof of the contraction mapping theorem gives information about how close a point x is to the fixed point of a contraction. We make use of the following general result. Lemma 6. Let X be a metric space and φ : X X be a contraction by c with < c < 1. Let y X be a (necessarily unique) fixed point of φ and let x X be arbitrary. Then d(y, x ) 1 1 c d(x, x 1 ). Proof. Extend the definition of x inductively by defining x i+1 = φ(x i ) for each integer i. Recall that {x i } tends to the fixed point y. Then, by the triangle inequality, j 1 d(x, y) = lim d(x, x j ) lim d(x i, x i+1 ) = j j Now observe that d(x i, x i+1 ) c i d(x, x 1 ). We conclude that d(x, y) d(x, x 1 ) c i = 1 1 c d(x, x 1 ). i= i= d(x i, x i+1 ). Proposition 7. The function g : V B r (a) which sends y to the fixed point of φ y is continuous. In fact, g(y 1 ) g(y ) A 1 y 1 y, for each y 1, y V. Proof. The second statement directly implies the continuity of g. Choose y 1, y V so that Set x 1 = g(y 1 ) and x = g(y ). We claim x 1 x < ɛ. Observe that i= φ y1 (x ) x = A 1( y 1 f(x ) ) = A 1 (y 1 y ). Thus, φ y1 (x ) x A 1 y 1 y. Recall x 1 is the fixed point of φ y1, which contracts distances by 1. Thus by the lemma above, x 1 x A 1 y 1 y = A 1 y 1 y.

5 THE INVERSE FUNCTION THEOREM 5 Now we address differentiability. Note that it suffices to prove that Dg(b) = Df(a) 1. This is because to check this at a point f(x) b, we can repeat the argument above to obtain a local inverse defined in a neighborhood of f(x). Since the inverses must agree on their intersection, the same argument implies that Dg ( f(x) ) = Df(x) 1. From this observation, the following Lemma completes the proof: Lemma 8. The function g is differentiable at b and Dg(b) = A 1. Proof. We will verify the definition of differentiable. We must show that g(y) [a + A 1 (y b)] lim =. y b y b To verify this, it suffices to show that for any ɛ > there is a c with < c < 1 so that for any point y in the neighborhood V = V (c) defined in Proposition 4, we have g(y) [a + A 1 (y b)] (3) < ɛ. y b Given ɛ >, we choose c so that c A 1 < ɛ and < c 1. Let r = r(c) and V = V (c). Choose y V c and let x = g(y) B r (a). As in the prior proof, we observe φ b (x) x = A 1( b f(x) ) = A 1 (b y). Observe that this relates to the numerator of the expression in equation 3: g(y) [a + A 1 (y b)] = x + A 1 (b y) a = φ b (x) a. Now recall that φ b contracts distances in B r (a) by a factor of c and fixes a, so g(y) [a + A 1 (y b)] = φb (x) a c x a. In summary, for y V, we have x = g(y) B r (a) and g(y) [a + A 1 (y b)] c x a c g(y) g(b) =. y a y b y b Then by Proposition 7, we have g(y) g(b) c A 1. Thus, g(y) [a + A 1 (y b)] c A 1 y b = c A 1 < ɛ. y a y b

d(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N

d(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

Metric Spaces and Topology

Metric 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 information

Review of Multi-Calculus (Study Guide for Spivak s CHAPTER ONE TO THREE)

Review of Multi-Calculus (Study Guide for Spivak s CHAPTER ONE TO THREE) Review of Multi-Calculus (Study Guide for Spivak s CHPTER ONE TO THREE) This material is for June 9 to 16 (Monday to Monday) Chapter I: Functions on R n Dot product and norm for vectors in R n : Let X

More information

INVERSE FUNCTION THEOREM and SURFACES IN R n

INVERSE FUNCTION THEOREM and SURFACES IN R n INVERSE FUNCTION THEOREM and SURFACES IN R n Let f C k (U; R n ), with U R n open. Assume df(a) GL(R n ), where a U. The Inverse Function Theorem says there is an open neighborhood V U of a in R n so that

More information

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

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,

More information

COMPLETE METRIC SPACES AND THE CONTRACTION MAPPING THEOREM

COMPLETE METRIC SPACES AND THE CONTRACTION MAPPING THEOREM COMPLETE METRIC SPACES AND THE CONTRACTION MAPPING THEOREM A metric space (M, d) is a set M with a metric d(x, y), x, y M that has the properties d(x, y) = d(y, x), x, y M d(x, y) d(x, z) + d(z, y), x,

More information

Economics 204 Fall 2011 Problem Set 2 Suggested Solutions

Economics 204 Fall 2011 Problem Set 2 Suggested Solutions Economics 24 Fall 211 Problem Set 2 Suggested Solutions 1. Determine whether the following sets are open, closed, both or neither under the topology induced by the usual metric. (Hint: think about limit

More information

Course 212: Academic Year Section 1: Metric Spaces

Course 212: Academic Year Section 1: Metric Spaces Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........

More information

Math 328 Course Notes

Math 328 Course Notes Math 328 Course Notes Ian Robertson March 3, 2006 3 Properties of C[0, 1]: Sup-norm and Completeness In this chapter we are going to examine the vector space of all continuous functions defined on the

More information

converges as well if x < 1. 1 x n x n 1 1 = 2 a nx n

converges 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 information

THE INVERSE FUNCTION THEOREM FOR LIPSCHITZ MAPS

THE INVERSE FUNCTION THEOREM FOR LIPSCHITZ MAPS THE INVERSE FUNCTION THEOREM FOR LIPSCHITZ MAPS RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA HOWARD@MATH.SC.EDU Abstract. This is an edited version of a

More information

4 Inverse function theorem

4 Inverse function theorem Tel Aviv University, 2014/15 Analysis-III,IV 71 4 Inverse function theorem 4a What is the problem................ 71 4b Simple observations before the theorem..... 72 4c The theorem.....................

More information

Mathematical Analysis Outline. William G. Faris

Mathematical Analysis Outline. William G. Faris Mathematical Analysis Outline William G. Faris January 8, 2007 2 Chapter 1 Metric spaces and continuous maps 1.1 Metric spaces A metric space is a set X together with a real distance function (x, x ) d(x,

More information

Analysis III Theorems, Propositions & Lemmas... Oh My!

Analysis III Theorems, Propositions & Lemmas... Oh My! Analysis III Theorems, Propositions & Lemmas... Oh My! Rob Gibson October 25, 2010 Proposition 1. If x = (x 1, x 2,...), y = (y 1, y 2,...), then is a distance. ( d(x, y) = x k y k p Proposition 2. In

More information

Definition 6.1. A metric space (X, d) is complete if every Cauchy sequence tends to a limit in X.

Definition 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

Implicit Functions, Curves and Surfaces

Implicit Functions, Curves and Surfaces Chapter 11 Implicit Functions, Curves and Surfaces 11.1 Implicit Function Theorem Motivation. In many problems, objects or quantities of interest can only be described indirectly or implicitly. It is then

More information

Analysis Finite and Infinite Sets The Real Numbers The Cantor Set

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

More information

MATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous:

MATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous: MATH 51H Section 4 October 16, 2015 1 Continuity Recall what it means for a function between metric spaces to be continuous: Definition. Let (X, d X ), (Y, d Y ) be metric spaces. A function f : X Y is

More information

Introduction to Topology

Introduction to Topology Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about

More information

Lecture Notes on Metric Spaces

Lecture Notes on Metric Spaces Lecture Notes on Metric Spaces Math 117: Summer 2007 John Douglas Moore Our goal of these notes is to explain a few facts regarding metric spaces not included in the first few chapters of the text [1],

More information

Many of you got these steps reversed or otherwise out of order.

Many of you got these steps reversed or otherwise out of order. Problem 1. Let (X, d X ) and (Y, d Y ) be metric spaces. Suppose that there is a bijection f : X Y such that for all x 1, x 2 X. 1 10 d X(x 1, x 2 ) d Y (f(x 1 ), f(x 2 )) 10d X (x 1, x 2 ) Show that if

More information

Nonlinear equations. Norms for R n. Convergence orders for iterative methods

Nonlinear equations. Norms for R n. Convergence orders for iterative methods Nonlinear equations Norms for R n Assume that X is a vector space. A norm is a mapping X R with x such that for all x, y X, α R x = = x = αx = α x x + y x + y We define the following norms on the vector

More information

MATH 202B - Problem Set 5

MATH 202B - Problem Set 5 MATH 202B - Problem Set 5 Walid Krichene (23265217) March 6, 2013 (5.1) Show that there exists a continuous function F : [0, 1] R which is monotonic on no interval of positive length. proof We know there

More information

f(x) f(z) c x z > 0 1

f(x) f(z) c x z > 0 1 INVERSE AND IMPLICIT FUNCTION THEOREMS I use df x for the linear transformation that is the differential of f at x.. INVERSE FUNCTION THEOREM Definition. Suppose S R n is open, a S, and f : S R n is a

More information

The Inverse Function Theorem 1

The Inverse Function Theorem 1 John Nachbar Washington University April 11, 2014 1 Overview. The Inverse Function Theorem 1 If a function f : R R is C 1 and if its derivative is strictly positive at some x R, then, by continuity of

More information

Math 140A - Fall Final Exam

Math 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 information

X. Linearization and Newton s Method

X. Linearization and Newton s Method 163 X. Linearization and Newton s Method ** linearization ** X, Y nls s, f : G X Y. Given y Y, find z G s.t. fz = y. Since there is no assumption about f being linear, we might as well assume that y =.

More information

SMSTC (2017/18) Geometry and Topology 2.

SMSTC (2017/18) Geometry and Topology 2. SMSTC (2017/18) Geometry and Topology 2 Lecture 1: Differentiable Functions and Manifolds in R n Lecturer: Diletta Martinelli (Notes by Bernd Schroers) a wwwsmstcacuk 11 General remarks In this lecture

More information

Math 421, Homework #9 Solutions

Math 421, Homework #9 Solutions Math 41, Homework #9 Solutions (1) (a) A set E R n is said to be path connected if for any pair of points x E and y E there exists a continuous function γ : [0, 1] R n satisfying γ(0) = x, γ(1) = y, and

More information

Most Continuous Functions are Nowhere Differentiable

Most Continuous Functions are Nowhere Differentiable Most Continuous Functions are Nowhere Differentiable Spring 2004 The Space of Continuous Functions Let K = [0, 1] and let C(K) be the set of all continuous functions f : K R. Definition 1 For f C(K) we

More information

1. Bounded linear maps. A linear map T : E F of real Banach

1. Bounded linear maps. A linear map T : E F of real Banach DIFFERENTIABLE MAPS 1. Bounded linear maps. A linear map T : E F of real Banach spaces E, F is bounded if M > 0 so that for all v E: T v M v. If v r T v C for some positive constants r, C, then T is bounded:

More information

MAS331: Metric Spaces Problems on Chapter 1

MAS331: Metric Spaces Problems on Chapter 1 MAS331: Metric Spaces Problems on Chapter 1 1. In R 3, find d 1 ((3, 1, 4), (2, 7, 1)), d 2 ((3, 1, 4), (2, 7, 1)) and d ((3, 1, 4), (2, 7, 1)). 2. In R 4, show that d 1 ((4, 4, 4, 6), (0, 0, 0, 0)) =

More information

Chapter 3. Differentiable Mappings. 1. Differentiable Mappings

Chapter 3. Differentiable Mappings. 1. Differentiable Mappings Chapter 3 Differentiable Mappings 1 Differentiable Mappings Let V and W be two linear spaces over IR A mapping L from V to W is called a linear mapping if L(u + v) = Lu + Lv for all u, v V and L(λv) =

More information

5 Integration with Differential Forms The Poincare Lemma Proper Maps and Degree Topological Invariance of Degree...

5 Integration with Differential Forms The Poincare Lemma Proper Maps and Degree Topological Invariance of Degree... Contents 1 Review of Topology 3 1.1 Metric Spaces............................... 3 1.2 Open and Closed Sets.......................... 3 1.3 Metrics on R n............................... 4 1.4 Continuity.................................

More information

The Heine-Borel and Arzela-Ascoli Theorems

The Heine-Borel and Arzela-Ascoli Theorems The Heine-Borel and Arzela-Ascoli Theorems David Jekel October 29, 2016 This paper explains two important results about compactness, the Heine- Borel theorem and the Arzela-Ascoli theorem. We prove them

More information

Real Analysis. Joe Patten August 12, 2018

Real Analysis. Joe Patten August 12, 2018 Real Analysis Joe Patten August 12, 2018 1 Relations and Functions 1.1 Relations A (binary) relation, R, from set A to set B is a subset of A B. Since R is a subset of A B, it is a set of ordered pairs.

More information

Topological properties

Topological 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

Course Summary Math 211

Course Summary Math 211 Course Summary Math 211 table of contents I. Functions of several variables. II. R n. III. Derivatives. IV. Taylor s Theorem. V. Differential Geometry. VI. Applications. 1. Best affine approximations.

More information

Contraction Mappings Consider the equation

Contraction Mappings Consider the equation Contraction Mappings Consider the equation x = cos x. If we plot the graphs of y = cos x and y = x, we see that they intersect at a unique point for x 0.7. This point is called a fixed point of the function

More information

Honours Analysis III

Honours Analysis III Honours Analysis III Math 354 Prof. Dmitry Jacobson Notes Taken By: R. Gibson Fall 2010 1 Contents 1 Overview 3 1.1 p-adic Distance............................................ 4 2 Introduction 5 2.1 Normed

More information

Math 209B Homework 2

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

More information

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real 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 information

Several variables. x 1 x 2. x n

Several variables. x 1 x 2. x n Several variables Often we have not only one, but several variables in a problem The issues that come up are somewhat more complex than for one variable Let us first start with vector spaces and linear

More information

Continuous Functions on Metric Spaces

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

More information

Differentiation in higher dimensions

Differentiation in higher dimensions Capter 2 Differentiation in iger dimensions 2.1 Te Total Derivative Recall tat if f : R R is a 1-variable function, and a R, we say tat f is differentiable at x = a if and only if te ratio f(a+) f(a) tends

More information

Definitions & Theorems

Definitions & Theorems Definitions & Theorems Math 147, Fall 2009 December 19, 2010 Contents 1 Logic 2 1.1 Sets.................................................. 2 1.2 The Peano axioms..........................................

More information

Exam 2 extra practice problems

Exam 2 extra practice problems Exam 2 extra practice problems (1) If (X, d) is connected and f : X R is a continuous function such that f(x) = 1 for all x X, show that f must be constant. Solution: Since f(x) = 1 for every x X, either

More information

Analysis II: The Implicit and Inverse Function Theorems

Analysis II: The Implicit and Inverse Function Theorems Analysis II: The Implicit and Inverse Function Theorems Jesse Ratzkin November 17, 2009 Let f : R n R m be C 1. When is the zero set Z = {x R n : f(x) = 0} the graph of another function? When is Z nicely

More information

7. Let X be a (general, abstract) metric space which is sequentially compact. Prove X must be complete.

7. Let X be a (general, abstract) metric space which is sequentially compact. Prove X must be complete. Math 411 problems The following are some practice problems for Math 411. Many are meant to challenge rather that be solved right away. Some could be discussed in class, and some are similar to hard exam

More information

NORMS ON SPACE OF MATRICES

NORMS ON SPACE OF MATRICES NORMS ON SPACE OF MATRICES. Operator Norms on Space of linear maps Let A be an n n real matrix and x 0 be a vector in R n. We would like to use the Picard iteration method to solve for the following system

More information

Metric spaces and metrizability

Metric spaces and metrizability 1 Motivation Metric spaces and metrizability By this point in the course, this section should not need much in the way of motivation. From the very beginning, we have talked about R n usual and how relatively

More information

g 2 (x) (1/3)M 1 = (1/3)(2/3)M.

g 2 (x) (1/3)M 1 = (1/3)(2/3)M. COMPACTNESS If C R n is closed and bounded, then by B-W it is sequentially compact: any sequence of points in C has a subsequence converging to a point in C Conversely, any sequentially compact C R n is

More information

Math 361: Homework 1 Solutions

Math 361: Homework 1 Solutions January 3, 4 Math 36: Homework Solutions. We say that two norms and on a vector space V are equivalent or comparable if the topology they define on V are the same, i.e., for any sequence of vectors {x

More information

Math General Topology Fall 2012 Homework 13 Solutions

Math General Topology Fall 2012 Homework 13 Solutions Math 535 - General Topology Fall 2012 Homework 13 Solutions Note: In this problem set, function spaces are endowed with the compact-open topology unless otherwise noted. Problem 1. Let X be a compact topological

More information

HOMEWORK FOR , FALL 2007 ASSIGNMENT 1 SOLUTIONS. T = sup T(x) Ax A := sup

HOMEWORK FOR , FALL 2007 ASSIGNMENT 1 SOLUTIONS. T = sup T(x) Ax A := sup HOMEWORK FOR 18.101, FALL 007 ASSIGNMENT 1 SOLUTIONS (1) Given a linear map T : R m R n Define the operator norm of T as follows: T(x) T := sup x 0 x Similarly, if A is a matrix, define the operator norm

More information

CHAPTER 9. Embedding theorems

CHAPTER 9. Embedding theorems CHAPTER 9 Embedding theorems In this chapter we will describe a general method for attacking embedding problems. We will establish several results but, as the main final result, we state here the following:

More information

Lecture 6: Contraction mapping, inverse and implicit function theorems

Lecture 6: Contraction mapping, inverse and implicit function theorems Lecture 6: Contraction mapping, inverse and implicit function theorems 1 The contraction mapping theorem De nition 11 Let X be a metric space, with metric d If f : X! X and if there is a number 2 (0; 1)

More information

Immerse Metric Space Homework

Immerse Metric Space Homework Immerse Metric Space Homework (Exercises -2). In R n, define d(x, y) = x y +... + x n y n. Show that d is a metric that induces the usual topology. Sketch the basis elements when n = 2. Solution: Steps

More information

08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms

08a. 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 information

Sobolev spaces. May 18

Sobolev spaces. May 18 Sobolev spaces May 18 2015 1 Weak derivatives The purpose of these notes is to give a very basic introduction to Sobolev spaces. More extensive treatments can e.g. be found in the classical references

More information

MASTERS EXAMINATION IN MATHEMATICS SOLUTIONS

MASTERS EXAMINATION IN MATHEMATICS SOLUTIONS MASTERS EXAMINATION IN MATHEMATICS PURE MATHEMATICS OPTION SPRING 010 SOLUTIONS Algebra A1. Let F be a finite field. Prove that F [x] contains infinitely many prime ideals. Solution: The ring F [x] of

More information

Lecture 5 - Hausdorff and Gromov-Hausdorff Distance

Lecture 5 - Hausdorff and Gromov-Hausdorff Distance Lecture 5 - Hausdorff and Gromov-Hausdorff Distance August 1, 2011 1 Definition and Basic Properties Given a metric space X, the set of closed sets of X supports a metric, the Hausdorff metric. If A is

More information

Lecture 9 Metric spaces. The contraction fixed point theorem. The implicit function theorem. The existence of solutions to differenti. equations.

Lecture 9 Metric spaces. The contraction fixed point theorem. The implicit function theorem. The existence of solutions to differenti. equations. Lecture 9 Metric spaces. The contraction fixed point theorem. The implicit function theorem. The existence of solutions to differential equations. 1 Metric spaces 2 Completeness and completion. 3 The contraction

More information

Inverse Function Theorem

Inverse Function Theorem Inverse Function Theorem Ethan Y. Jaffe 1 Motivation When as an undergraduate I first learned the inverse function theorem, I was using a textbook of Munkres [1]. The proof presented there was quite complicated

More information

Maths 212: Homework Solutions

Maths 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 information

Math 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 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 information

Part A. Metric and Topological Spaces

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

More information

FIXED POINT ITERATIONS

FIXED POINT ITERATIONS FIXED POINT ITERATIONS MARKUS GRASMAIR 1. Fixed Point Iteration for Non-linear Equations Our goal is the solution of an equation (1) F (x) = 0, where F : R n R n is a continuous vector valued mapping in

More information

M311 Functions of Several Variables. CHAPTER 1. Continuity CHAPTER 2. The Bolzano Weierstrass Theorem and Compact Sets CHAPTER 3.

M311 Functions of Several Variables. CHAPTER 1. Continuity CHAPTER 2. The Bolzano Weierstrass Theorem and Compact Sets CHAPTER 3. M311 Functions of Several Variables 2006 CHAPTER 1. Continuity CHAPTER 2. The Bolzano Weierstrass Theorem and Compact Sets CHAPTER 3. Differentiability 1 2 CHAPTER 1. Continuity If (a, b) R 2 then we write

More information

Metric Spaces Math 413 Honors Project

Metric Spaces Math 413 Honors Project Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X X R such that for all x, y, z X: i) d(x, y) = d(y, x); ii) d(x, y) = 0 if and only

More information

B. Appendix B. Topological vector spaces

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

More information

Notes on uniform convergence

Notes on uniform convergence Notes on uniform convergence Erik Wahlén erik.wahlen@math.lu.se January 17, 2012 1 Numerical sequences We begin by recalling some properties of numerical sequences. By a numerical sequence we simply mean

More information

Chapter 2. Metric Spaces. 2.1 Metric Spaces

Chapter 2. Metric Spaces. 2.1 Metric Spaces Chapter 2 Metric Spaces ddddddddddddddddddddddddd ddddddd dd ddd A metric space is a mathematical object in which the distance between two points is meaningful. Metric spaces constitute an important class

More information

Real Analysis Notes. Thomas Goller

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..............................

More information

Richard S. Palais Department of Mathematics Brandeis University Waltham, MA The Magic of Iteration

Richard S. Palais Department of Mathematics Brandeis University Waltham, MA The Magic of Iteration Richard S. Palais Department of Mathematics Brandeis University Waltham, MA 02254-9110 The Magic of Iteration Section 1 The subject of these notes is one of my favorites in all mathematics, and it s not

More information

FIXED POINT METHODS IN NONLINEAR ANALYSIS

FIXED POINT METHODS IN NONLINEAR ANALYSIS FIXED POINT METHODS IN NONLINEAR ANALYSIS ZACHARY SMITH Abstract. In this paper we present a selection of fixed point theorems with applications in nonlinear analysis. We begin with the Banach fixed point

More information

Lecture 11 Hyperbolicity.

Lecture 11 Hyperbolicity. Lecture 11 Hyperbolicity. 1 C 0 linearization near a hyperbolic point 2 invariant manifolds Hyperbolic linear maps. Let E be a Banach space. A linear map A : E E is called hyperbolic if we can find closed

More information

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................

More information

Chapter 2 Metric Spaces

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

More information

2 Metric Spaces Definitions Exotic Examples... 3

2 Metric Spaces Definitions Exotic Examples... 3 Contents 1 Vector Spaces and Norms 1 2 Metric Spaces 2 2.1 Definitions.......................................... 2 2.2 Exotic Examples...................................... 3 3 Topologies 4 3.1 Open Sets..........................................

More information

MATH 411 NOTES (UNDER CONSTRUCTION)

MATH 411 NOTES (UNDER CONSTRUCTION) MATH 411 NOTES (NDE CONSTCTION 1. Notes on compact sets. This is similar to ideas you learned in Math 410, except open sets had not yet been defined. Definition 1.1. K n is compact if for every covering

More information

MA651 Topology. Lecture 10. Metric Spaces.

MA651 Topology. Lecture 10. Metric Spaces. MA65 Topology. Lecture 0. Metric Spaces. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Linear Algebra and Analysis by Marc Zamansky

More information

SOLUTIONS TO SOME PROBLEMS

SOLUTIONS TO SOME PROBLEMS 23 FUNCTIONAL ANALYSIS Spring 23 SOLUTIONS TO SOME PROBLEMS Warning:These solutions may contain errors!! PREPARED BY SULEYMAN ULUSOY PROBLEM 1. Prove that a necessary and sufficient condition that the

More information

THE CONTRACTION MAPPING THEOREM

THE 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 information

1 Directional Derivatives and Differentiability

1 Directional Derivatives and Differentiability Wednesday, January 18, 2012 1 Directional Derivatives and Differentiability Let E R N, let f : E R and let x 0 E. Given a direction v R N, let L be the line through x 0 in the direction v, that is, L :=

More information

Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011

Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011 Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011 Section 2.6 (cont.) Properties of Real Functions Here we first study properties of functions from R to R, making use of the additional structure

More information

Topology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA address:

Topology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA  address: Topology Xiaolong Han Department of Mathematics, California State University, Northridge, CA 91330, USA E-mail address: Xiaolong.Han@csun.edu Remark. You are entitled to a reward of 1 point toward a homework

More information

Math 127C, Spring 2006 Final Exam Solutions. x 2 ), g(y 1, y 2 ) = ( y 1 y 2, y1 2 + y2) 2. (g f) (0) = g (f(0))f (0).

Math 127C, Spring 2006 Final Exam Solutions. x 2 ), g(y 1, y 2 ) = ( y 1 y 2, y1 2 + y2) 2. (g f) (0) = g (f(0))f (0). Math 27C, Spring 26 Final Exam Solutions. Define f : R 2 R 2 and g : R 2 R 2 by f(x, x 2 (sin x 2 x, e x x 2, g(y, y 2 ( y y 2, y 2 + y2 2. Use the chain rule to compute the matrix of (g f (,. By the chain

More information

Course 224: Geometry - Continuity and Differentiability

Course 224: Geometry - Continuity and Differentiability Course 224: Geometry - Continuity and Differentiability Lecture Notes by Prof. David Simms L A TEXed by Chris Blair 1 Continuity Consider M, N finite dimensional real or complex vector spaces. What does

More information

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9 MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended

More information

MAT 473 Intermediate Real Analysis II

MAT 473 Intermediate Real Analysis II MAT 473 Intermediate Real Analysis II John Quigg Spring 2009 revised February 5, 2009 Derivatives Here our purpose is to give a rigorous foundation of the principles of differentiation in R n. Much of

More information

Continuity. Chapter 4

Continuity. Chapter 4 Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

More information

Commutative Banach algebras 79

Commutative Banach algebras 79 8. Commutative Banach algebras In this chapter, we analyze commutative Banach algebras in greater detail. So we always assume that xy = yx for all x, y A here. Definition 8.1. Let A be a (commutative)

More information

The Contraction Mapping Theorem and the Implicit and Inverse Function Theorems

The Contraction Mapping Theorem and the Implicit and Inverse Function Theorems The Contraction Mapping Theorem and the Implicit and Inverse Function Theorems The Contraction Mapping Theorem Theorem The Contraction Mapping Theorem) Let B a = { x IR d x < a } denote the open ball of

More information

Half of Final Exam Name: Practice Problems October 28, 2014

Half of Final Exam Name: Practice Problems October 28, 2014 Math 54. Treibergs Half of Final Exam Name: Practice Problems October 28, 24 Half of the final will be over material since the last midterm exam, such as the practice problems given here. The other half

More information

2. Metric Spaces. 2.1 Definitions etc.

2. Metric Spaces. 2.1 Definitions etc. 2. Metric Spaces 2.1 Definitions etc. The procedure in Section for regarding R as a topological space may be generalized to many other sets in which there is some kind of distance (formally, sets with

More information

Stereographic projection and inverse geometry

Stereographic projection and inverse geometry Stereographic projection and inverse geometry The conformal property of stereographic projections can be established fairly efficiently using the concepts and methods of inverse geometry. This topic is

More information

1 )(y 0) {1}. Thus, the total count of points in (F 1 (y)) is equal to deg y0

1 )(y 0) {1}. Thus, the total count of points in (F 1 (y)) is equal to deg y0 1. Classification of 1-manifolds Theorem 1.1. Let M be a connected 1 manifold. Then M is diffeomorphic either to [0, 1], [0, 1), (0, 1), or S 1. We know that none of these four manifolds are not diffeomorphic

More information

Lecture 4: Completion of a Metric Space

Lecture 4: Completion of a Metric Space 15 Lecture 4: Completion of a Metric Space Closure vs. Completeness. Recall the statement of Lemma??(b): A subspace M of a metric space X is closed if and only if every convergent sequence {x n } X satisfying

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets

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

More information