Mathematics for Economists Chapter 1: Basic DeÖnitions 1. Operations on sets Intersection X \ Y; union X [ Y; complement Z ! Z c ; di§erence X Y = X \ Y c : Distributivity X \ ( Y [ Z ) = ( X \ Y ) [ ( X \ Z ) X [ ( Y \ Z ) = ( X [ Y ) \ ( X [ Z ) ( X \ Y ) c = X c [ Y c ; ( X [ Y ) c = X c \ Y c union, intersection are commutative and associative Cartesian product: X Y is the set of ordered pairs ( x; y ) with x 2 X; y 2 Y: The product X X is not the same as the set of subsets of X with two elements. 2. Functions, mappings f : X ! Y has domain X and values/image in Y; it associates with every x 2 X an element y = f ( x ) in Y: The range of f is the set f ( X ) = f y 2 Y j 9 x 2 X : y = f ( x ) g The graph of f is the subset of X Y : graph ( f ) = f ( x; f ( x )) j x 2 X g It is sometime convenient to identify a function with its graph: then a func- tion is deÖned as a subset of X Y such that every x -section contains exactly one y 2 Y: The set of functions from X into Y is denoted Y X : Example: the set of subsets of X is denoted f 0 ; 1 g X or 2 X : The function f 2 Y X is onto Y if f ( X ) = Y ; it is one-to-one if x 6 = x 0 ) f ( x ) 6 = f ( x 0 ); it is a bijection if it is one to one and onto Y: In the later case, we can speak of the inverse of f; the function f 1 from Y into X deÖned by y = f ( x ) , x = f 1 ( y ) : Note: if f 2 Y X is not a bijection, its inverse as a function from Y to X is not deÖned. However, if f is one-to-one, we may identify it with a function g from X into f ( X ) ; and that function has an inverse g 1 from f ( X ) into X: Hence the need to always specify the domain of a function. Composition of f 2 Y X ; g 2 Z Y : g f ( x ) = g ( f ( x )) : For instance, if id X denotes the identity mapping of X; i.e., id X ( x ) = x for all x; and f 2 Y X is a bijection, we have f f 1 = id Y ; f 1 f = id X : The composition is an associative operation, it is not commutative. If f 2 Y X and g 2 X Y are such that g f = id X ; we cannot deduce g = f 1 ; or f = g 1 : If g f = id X and f g = id Y ; we can. Exercise 1 What can we deduce from g f = id X , or f g = id Y ? 1 Exercise 2 If f 2 Y X and g 2 Z Y are onto, so is g f: If they are one to one, so is g f: A bijection from X into itself is called a permutation . Their set is denoted A ( X ) : The permutations of X form a group for the composition: every f 2 A ( X ) has an inverse f 1 2 A ( X ) : Exercise 3 If X contains three or more distinct elements, construct f; g 2 A ( X ) such that f g 6 = g f: Exercise 4 If X is Önite, j X j = n; then A ( X ) is Önite and A ( X ) = n ! = n ( n 1) ::: 2 : 1 : Exercise 5 A permutation h is called elementary if it exchanges two elements x; y of A and leaves every other element Öxed: h ( x ) = y; h ( y ) = x and h ( z ) = z for all z 6 = x; y: If j X j is Önite, every permutation f 2 A ( X ) can be written as a product of elementary permutations. Exercise 6 If X = f 1 ; 2 ; :::; n g and a permutation f of X is written as f i 1 ; i 2 ; :::; i n g ; where i k = f ( k ) for k = 1 ; :::; n; we can check the parity of f as follows. Denote sign ( z ) = +1 ( resp: 1 ; resp: 0 ) if z is positive ( resp: negative, resp: zero). Then deÖne the signature of a permutation f as ( f ) = 1 k;l n k<l sign ( i l i k ) Compute the signature of the identity permutation and of an elementary permu- tation. Then show ( f g ) = ( f ) ( g ) for all permutations f,g. Deduce that if a permutation is the product of an even ( resp. odd) number of elementary permutations , then it can not be the product of an odd ( resp. even) number of such e. p.s. We can speak of a permutation being î oddî or î evenî . Show that A(X) is evenly split between these 2 types of permutations. Correspondences : F; with domain X and values in Y associates with every x 2 X a non-empty subset F ( x ) of Y: Notation: F: X Y: We use the notation insert to distinguish correspondences from functions. The graph of F is graph ( F ) = f ( x; y ) 2 X Y j y 2 F ( x ) g Alternative deÖnition of a correspondence: a subset of X Y such that every x -section is non empty. The range of F is F ( X ) = [ x 2 X F ( x ) Composition of F : X Y and G : Y Z : G F ( x ) = [ y 2 F ( x ) G ( y ) The composition is associative. 3. Binary relations A binary relation R on X; Y; is a subset of X Y: The conventional notation is: 2 xRy , def ( x; y ) 2 R For instance, a function f 2 Y X can be viewed as a binary relation. An equivalence relation on X is a binary relation R on X; X that is reá exive: xRx for all x 2 X symmetric: xRy , yRx for all x; y 2 X transitive: xRy and yRz , xRz for all x; y; z 2 X A partition of X is a family a subsets X t ; t 2 T (namely an element in (2 X ) T ) such that X t \ X t 0 = ; for all t; t 0 2 T and [ t 2 T X t = X: An equivalence relation on X determines a partition of X by its equiva- lence classes; the set of the equivalence classes is denoted X= : Conversely, a partition of X determines an equivalence relation. So the concepts of equiva- lence relations and of partition are interchangeable. Example: aRb i§ ì a lives within 100 miles of b î is not an e.r. aRb i§ ì a and b have the same blood typeî is an e.r. What about ì a and b sat in the same class at least onceî ? ì a and b have a common friendî ? Exercise 7 Counting the number p n of distinct partitions of a set X with n elements is not a simple matter. Show the recursive formula: p n ( k ) = p n 1 ( k ) k + p n 1 ( k 1) where p n ( k ) is the number of partitions with k elements in a n-set. Deduce p n for small n : p 2 = 2 ; p 3 = 5 ; p 4 = 15 ; p 5 =? ; p 6 =? A linear order is a binary relation R on X that is complete, transitive and antisymmetric: complete: xRy or yRx for all x; y 2 X: antisymmetric: xRy and yRx ) x = y; for all x; y: Conventional notation x y; larger than, preferred to, higher than; or x y; lower than, etc. When X is Önite,the set of linear orders can be identiÖed with A ( X ) and in particular its cardinality is n ! . This is not true when X is inÖnite. Preordering/Preference relation/rational preferences : a binary relation R on X that is complete, and transitive. Conventional notation x % y; greater than or equal to, preferred or indi§er- ent to, not lower than. 3 The indi§erence relation s associated with % is a x s y , def f x % y and y % x g : It is an equivalence relation on X: Every preordering % on X can be decomposed into i) the partition of its indi§erence classes and ii) a linear ordering on X= s ; the set of its indi§erence classes. Conversely, any pair of an equivalence relation s on X and a linear ordering on X= s deÖnes a unique preordering on X: Example: X R n ; pick a function f 2 R X and deÖne x % y by f ( x ) % f ( y ) : Then % is a preordering. (If % is a preference relation, f is a utility function representing % ) : If X is Önite, every preference relation on X can be represented by a ( by many) utility functions. This is not true when X is inÖnite. The simplest example is the lexicographic ordering . Example: X = R n ; x = ( x 1 ; :::; x n ) ; the lexicographic ordering of R n is deÖned by: x y , def f x 1 > y 1 g or f x 1 = y 1 ; and x 2 > y 2 g or ... or f x i = y i for i = 1 ; :::; n 1 and x n > y n g It is a linear ordering of R n : Exercise 8 Check that the recursive formula to count the number r n of di§ erent preorderings over a set X with n elements is : r n ( k ) = k ( r n 1 ( k )+ r n 1 ( k 1)) Compute r 2 ; r 3 ; r 4 ; r 5 : 4. Cardinals The binary relation over the set of all sets ( X; Y ) 2 R , def there exists f 2 Y X and onto Y , there exists g 2 X Y and one-to-one on Y is a preordering. Its indi§erence classes are written j X j and R is written j X j j Y j : We call j X j the cardinal of X: Also j X j > j Y j means j X j j Y j and Not j Y j j X j : The equivalence relation \ X and Y have the same cardinality " is character- ized as follows: j X j = j Y j , there is a one-to-one f 2 Y X and a one-to-one g 2 X Y , there is an onto f 2 Y X and an onto g 2 X Y , there is a bijection h from X into Y: Note: the latter equivalence requires a non trivial proof, given in the Appen- dix. If X Y; then j X j j Y j : A set X is Önite i§ for any proper subset X 0 of X , we have j X 0 j < j X j : It is inÖnite i§ for some proper subset X 0 of X; we have j X 0 j = j X j (note that j X 0 j j X j is clear, if X 0 is a subset of X ) : 4 The set N of natural integers is the set of Önite cardinals. It is an inÖnite set. Conventional notation N = f 0 ; 1 ; 2 ; :::; n; ::: g : The preordering of Önite cardinals is called the natural order of N : Note that j N j is the smallest inÖnite cardinal. A set with cardinality j N j is called enumerable Examples : Q , the set of rational numbers, is enumerable. R ; the set of real numbers, is not: j R j > j N j : (this important fact is an easy consequence of the decimal development of real numbers: see Chapter 2). N = N ::: N | {z } k tim es is enumerable: N k = j N j : Continuum hypothesis: there is no cardinal between j R j and j N j : Exercise 9 R k = j R j Exercise 10 If X; Y are Önite: j X [ Y j = j X j + j Y j j X \ Y j and Y X = j Y j j X j Exercise 11 Show the following properties for intervals of real numbers: j [0 ; 1] j = j [0 ; 1[ j = j ]0 ; 1[ j Exercise 12 For any set, Önite or not, 2 X > j X j Exercise 13 The lexicographic ordering of R 2 cannot be represented by a utility function. Suppose such a function u ( x 1 ; x 2 ) exists, then we can pick for all x 1 2 R a rational number r ( x 1 ) such that u ( x 1 ; 2) > r ( x 1 ) > u ( x 1 ; 1) : Deduce a contradiction. Appendix: Two proofs on Cardinals 1. The relation j X j j Y j is complete. Given two sets X; Y; consider the set Z of triples z = ( A; B; f ) where A X; B Y; and f is a bijection from A into B The binary relation % on Z : ( A; B; f ) % ( A 0 ; B 0 ; f 0 ) , def A 0 A; B 0 B; f = f 0 on A 0 ; is clearly transitive (but not complete). Given any subset Z 0 of Z on which % is complete, we construct an upper bound e z of Z 0 (i.e., e z % z for all z 2 Z 0 ) as follows: e z = ( e A; e B; e f ) where e A = [ A; e B = [ B and e f = f on A for all z = ( A; B; f ) in Z 0 : An important result in set theory, known as Zorní s lemma, says that the relation % must have a maximal element z in Z; namely such that z z holds for no z 2 Z: Let z = ( A ; B ; f ) : If X A and Y B are both non empty, we can ìaugmentî z by adding a 2 X A to A ; b 2 Y B to B and setting f ( a ) = b: This contradicts the maximality of z : Therefore A = X or B = Y or both. The former implies j X j j Y j and the latter j Y j j X j : 5 2. If j X j = j Y j ; there exist a bijection from X into Y: a. We prove an auxiliary result Örst. Let A; A 1 ; A 2 be 3 nested sets, A 2 A 1 A; and f a bijection from A into A 2 : Then there exists a bijection g from A into A 1 : We construct a nested sequence A 2 A 3 A 4 ::: as follows: A 3 = f ( A 1 ) ( ) A 3 f ( A ) = A 2 ) A 4 = f ( A 2 ) ( ) A 4 f ( A 1 ) = A 3 ) A 5 = f ( A 3 ) ( ) A 5 f ( A 2 ) = A 4 ) etc. Note that f ( A A 1 ) = A 2 A 3 (because f is a bijection) and similarly f ( A 2 A 3 ) = A 4 A 5 (for the same reason) f ( A 4 A 5 ) = A 6 A 7 etc. DeÖne g 2 A A g = f on A A 1 ; A 2 A 3 ; A 4 A 5 ; ::: g = identity on A 1 A 2 ; A 3 A 4 ; A 5 A 6 ; ::: Write B = ( A A 1 ) [ ( A 2 A 3 ) [ ( A 4 A 5 ) [ ::: C = ( A 1 A 2 ) [ ( A 3 A 4 ) [ ( A 5 A 6 ) [ ::: and note that f ( B ) B ; id ( C ) = C: Therefore g is one-to-one on A: More- over f ( B ) [ C = A 1 ; therefore g is a bijection from A into A 1 : b. We Öx X; Y such that j X j j Y j and j Y j j X j : Thus there are two one- to-one mappings f; g; where f is from X into Y; and g from Y into X: DeÖne A = X; A 1 = g ( Y ) ; A 2 = ( g f )( X ) : Note that g f is a bijection from A into A 2 ; and moreover A 2 A 1 A: By the auxiliary result, there is a bijection h from A into A 1 : Thus h 1 g is the desired bijection from Y into X: 6 Chapter 2: Real Analysis 1. Sequences Z = f :::; n; :::; 2 ; 1 ; 0 ; 1 ; 2 ; :::; n; ::: g is the set of relative integers. Q = f p q ; p; q 2 Z g is the set of rational numbers. R =] 1 ; + 1 [ is the set of real numbers R + = [0 ; + 1 [ is the set of non negative real numbers R ++ =]0 ; + 1 [ is the set of positive real numbers Integer part: for all x 2 R ; this is the unique relative integer p; p 2 Z ; such that p x < p + 1; it is written b x c R is archimedian: 8 b 2 R 9 p 2 N p > a Exercise 14 8 a > 0 ; a 2 R ; 8 b 2 R 9 p 2 N p:a > b Exercise 15 8 " > 0 ; " 2 R ; 9 2 Q : 0 < < " A sequence of real numbers is an element of R N ; and the conventional no- tation is ( x n ) n 2 N or ( x 0 ; x 1 ; :::; x n ; ::: ) or ( x 1 ; :::; x n ; ::: ) : Note: a sequence in an arbitrary set X is an element of X N : The sequence ( x n ) is bounded if there is a 2 N such that j x n j a for all n: The sequence ( x n ) of real numbers converges to x if we have 8 " > 0 9 m 2 N 8 n m j x n x j " a property that we write lim n x n = x Note: in the deÖnition above, we can limit the choice of " to the numbers 1 p ; p = 1 ; 2 ::: Thus, for instance, lim n !1 x n = x is equivalent to 8 p 2 N 9 m 2 N 8 n m j x n x j 1 p Decimal development of a real number. Given x 2 R ; we construct inductively a sequence ( a n ) of integers such that 0 a n 9 for all n b 10 x c = 10 b x c + a 1 b 10 2 x c = 10 2 b x c + 10 a 1 + a 2 ... b 10 n x c = 10 n b x c + 10 n 1 a 1 + 10 n 2 a 2 + ::: 10 a n 1 + a n i.e., for all n the decimal development of the integer b 10 n x c 10 n b x c is a 1 ; a 2 ; :::; a n : The sequence x n = b x c + n P k =1 10 k a k converges to x Conversely if the sequence ( a n ) of integers between 0 on 9 is such that: lim f n b x c + n P k =1 10 k a k g = x then this sequence is the decimal development of x; unless a n = 9 for all n large enough. A number is rational if andf only if its decimal development is periodic. Exercise 16 Prove the two statements above. 7 Two important applications of the decimal development : Between any two distinct real numbers, there is an inÖnityof rational num- bers and an inÖnity of irrational numbers. j R j > j N j : Operations on limits: lim n x n + lim n y n = lim n ( x n + y n ) (lim n x n ) (lim n y n ) = lim n ( x n y n ) f lim n x n 6 = 0 g ) lim n ( 1 x n ) = 1 lim n x n where we assume that the left-hand limits exist, and deduce that the right- hand limit exists, too. f ( x n ) bounded and lim n y n = 0 g ) lim n x n y n = 0 Exercise 17 The sum (or product) of two non convergent sequences may be convergent. Exercise 18 The product of a bounded sequence and a convergent (resp. a non convergent) sequence, may or may not be convergent. Cauchy sequences: A Cauchy sequence is a sequence (x n ) such that : 8 n x n y n ) lim n x n lim n y n Cauchy sequences: A Cauchy sequence is a sequence (x n ) such that : 8 " > 0 9 m 2 N 8 n; n 0 m j x n x n 0 j " Fundamental result: a sequence of real numbers is convergent if and only if it is a Cauchy sequence. Note that the result is in a way tautological, because the set of real numbers is deÖned as the set of Cauchy sequences in Q , up to a quotient operation. In Q , a Cauchy sequence may not be convergent. Example: a non periodic decimal development. Exercise 19 Prove this claim. Upper and lower bounds: the subset X of R is bounded above (resp. below ) if there exists an integer a such that 8 x 2 X; x a (resp. x a ) : In this case a is called an upper bound of X (resp. a lower bound ). A key application of the fundamental result : every set X bounded above has a smallest upperbound denoted sup( X ) and called the supremum of X . Every set X bounded below has a largest lowerbound inf( X ) called the inÖmum of X 8 Exercise 20 The supremum a = sup( X ) is characterized by the two following properties: a is an upper bound of X 8 " > 0 9 x 2 X : x > a " (a similar statement holds for the inÖmum). The sequence ( x n ) is called non-decreasing (resp. non increasing) if x n x n +1 for all n ( resp. x n x n +1 ) : A non decreasing (resp. non increasing) sequence that is bounded above (resp. below) converges to the supremum (resp. inÖmum) of its values. If I n is a nested sequence of closed intervals I n = [ a n ; b n ] ; i.e., I n +1 I n for all n; then the intersection \ n I n is non empty. If, moreover, lim n ( b n a n ) = 0 ; then \ n I n contains exactly one real number. The lim inf and lim sup of a bounded sequence ( x n ) are deÖned as follows: lim sup( x n ) = inf f sup m n x m j n = 1 ; 2 ; ::: g = lim n ( sup m n x m ) lim inf( x n ) = sup f inf m n x m j n = 1 ; 2 ; ::: g = lim n ( inf m n x m ) Fact: lim inf( x n ) lim sup( x n ) with equality if and only if ( x n ) converges, in which case its limit is the common value. Exercise 21 The lim sup and lim inf operators do not commute with addition, or with multiplication. Exercise 22 Prove : lim sup( x n + y n ) lim sup x n + lim sup y n If x n ; y n are positive prove: lim sup( x n y n ) (lim sup x n ) (lim sup y n ) Similar properties hold for lim inf : What is the composition of lim sup / lim inf with the inverse operator: x n ! 1 x n ? Limit points of a sequence ( x n ) : a number y such that 8 " > 0 8 m 2 N 9 n m j x n y j ": A subsequence of the sequence ( x n ) 2 R N is a sequence ( x ' ( n ) ) = ( x ' (0) ; x ' (1) ; :::; x ' ( n ) ; ::: ) where ' is a strictly increasing function of N into itself. Every subsequence of a converging sequence converges to the same limit. The number y is a limit point of the sequence ( x n ) if and only if there exists a subsequence ( x ' ( n ) ) of ( x n ) converging to y: If ( x n ) is a bounded sequence, lim n sup x n and lim n inf x n are, respectively, its largest and its smallest limit point. Corollary: a bounded sequence is convergent if and only if it has a unique limit point. Corollary: every bounded sequence has at least one limit point. Note: The set of limit points of a sequence ( x n ) can be any enumerable subset of R : Given a Önite set f 0 ; :::; K 1 g the sequence x n = k if k is the remainder of ( n : K ) has the limit points 0 ; :::; K 1 and only those. 9 Exercise 23 Given an arbitrary sequence ( n ) of real numbers, construct a sequence ( x n ) of which the limit points are exactly the values of the sequence ( n ) and its limit points. 2. Continuity Interval: an interval of R is a subset I such that 8 x; y = f x 2 I and y 2 I g ) [ x; y ] I There are nine types of interval depending upon their endpoints being real numbers or 1 ; and on whether an endpoint belongs to I or not. Limit of a function over a subset: Let A be a subset of R and a be in the closure A of A (that is, a is the limit of at least one sequence in A ), we say that f 2 R A converges to b 2 R when x converges to a in A (or simply, the limit of f in A at a is b ) and we write lim x ! a x 2 A f ( x ) = b; the following property: 8 " > 0 9 > 0 8 x 2 A : j x a j ) j f ( x ) b j " We say that the limit of f in A at a is + 1 if we have: 8 n 2 N 9 > 0 8 x 2 A : j x a j ) f ( x ) n We say that the limit of f in A at + 1 is b if we have: 8 " > 0 9 n 2 N 8 x 2 A x n ) j f ( x ) b j " Exercise 24 DeÖne similarly lim x ! a x 2 A f ( x ) = 1 ; lim x ! 1 x 2 A f ( x ) = + 1 ; etc. Equivalent formulation: for any sequence ( x n ) in A : lim n x n = a ) lim n f ( x n ) = b Examples: lim x ! 0 x> 0 Logx = 1 ; lim x ! 2 x< 2 b x c = 1; lim x ! 2 x> 2 b x c = 2 If f is monotone (non increasing or non decreasing) then f has a right-limit and a left-limit (in A ) everywhere: lim x ! a x<a x 2 A f ( x ) and lim x ! a x>a x 2 A f ( x ) exist for all a 2 A (in the above statement, the limit may be 1 ; and a may be replaced by 1 ) : Continuity of a function f 2 R A where A is a bounded interval: [ ; ] ; ] ; ] ; [ ; [ ; ] ; [ : If a is interior to A : f is continuous at a def , lim x ! a f ( x ) = f ( a ) , lim n f ( x n ) = f ( a ) for every ( x n ) s.t. lim n x n = a if a = 2 A : f is continuous at a def , lim x ! x> f ( x ) = f ( a ) , lim n f ( x n ) = f ( a ) for every ( x n ) s.t. x n and lim n x n = a 10 Exercise 25 If a = 2 A write the similar deÖnition of lim n x n = a: Right continuity, left continuity Same premises as above. If a is interior to A : f is right-continuous at a def , lim x ! x> f ( x ) = f ( a ) f is left-continuous at a def , lim x ! x< f ( x ) = f ( a ) If a = : f is right-continuous at a , f is continuous at a left continuity is not deÖned. Operations on continuous functions: the sum, the product, the quotient respect the property of continuity at a given point, that of right continuity, and that of left continuity. The composition of functions respects the continuity property: assume f 2 R A ; g 2 R B ; f ( A ) B; then { f continuous at a; and g continuous at b = f ( a ) g ) g f continuous at a: Exercise 26 The composition of right-continuous functions may not be right- continuous. The supremum (resp. inÖmum) of Önitely many functions respects the prop- erties of continuity, right-continuity, left-continuity. f k 2 R A ; k = 1 ; :::; K ; g ( x ) = sup k =1 ;:::;K f k ( x ) f for all k; f k is continuous at a g ) f g is continuous at a g Examples: f ( x ) = b x c ; A = R ; is right-continuous everywhere and continuous for non integer values of x: f ( x ) = sin x if x 6 = 0; f (0) = 0; A = R is neither right nor left-continuous at 0; is continuous everywhere else. f ( x ) = 0 if x 2 Q ; f ( x ) = 1 if x 2 R Q ; is neither right nor left-continuous anywhere. If f is monotone on R ; it is continuous at all points, with the exception of at most an enumerable set of points. Exercise 27 Construct a monotone function on R that is discontinuous at every rational number and continuous at every irrational number. What about a function discontinuous at every irrational number and continuous at every rational number ? A closed bounded interval of R is a set [ a; b ] = f x j a x b g where a; b 2 R : If f is continuous on A (i.e., continuous at every point of A ) ; and [ a; b ] A; then f ([ a; b ]) is a closed bounded interval. This crucial observation has two important corollaries. 11 Intermediate value theorem: if f is continuous on the closed, bounded inter- val [ a; b ] ; f takes all the values between f ( a ) and f ( b ) : 8 x 2 ] f ( a ) ; f ( b )[ 9 c 2 ] a; b [ f ( c ) = x Maximum theorem: if f is continuous on the closed, bounded interval [ a; b ] ; f reaches its maximum and its minimum: 9 c; d 2 [ a; b ] : f ( c ) = sup a x b f ( x ); f ( d ) = inf a x b f ( x ) Variant of the IVT: if f is continuous on the interval I; then f ( I ) is an interval of R : Exercise 28 We may have I =] a; b [ and f ( I ) = [ c; d ] ; or I = [ a; + 1 [ and f ( I ) =] c; d [ ; etc. An application of the IVT: a polynomial of odd degree has at least one real root. Homeomorphism Let I be an interval and f 2 R I : The four following properties are equivalent: i) f is a bijection from I into f ( I ) , f is continuous on I and f 1 is continuous on f ( I ) ; ii) f is continuous and one-to-one on I; iii) f is continuous and strictly monotone (strictly increasing or strictly de- creasing), iv) f is strictly monotone and f ( I ) is an interval. Uniform Continuity: f 2 R A is uniformly continuous on A if we have: 8 " > 0 9 > 0 8 x; y 2 A : j x y j ) j f ( x ) f ( y ) j " Let I be a closed bounded interval and f 2 R I Then f f is continuous on I g , f f is uniformly continuous on I g : On the other hand, if A is not bounded, uniform continuity is a more de- manding property than continuity: f ( x ) = x 2 is c: but not u:c: on R + f ( x ) = sin x is u:c: on R Exercise 29 Show that f ( x ) = 1 x is uniformly continuous on [1 ; + 1 [ : Show that f ( x ) = sin 1 x is not uniformly continuous on ]0,1]. Is f ( x ) = sin( x 2 ) u.c. on R ? What about f ( x ) = sin( x 2 ) x for x 6 = 0 ; f (0) = 0? 3. Derivatives Given f 2 R A ; where A is an interval, and a an interior point of A; we say that f is di§ erentiable (or derivable ) at a if lim x ! a x 6 = a f ( x ) f ( a ) x a exists. This limit is called the derivative of f at a and denoted f 0 ( a ) : 12 Right-derivative of f at a : it is the limit lim x ! a x>a f ( x ) f ( a ) x a = f 0 + ( a ) ; when this limit exists Left-derivative of f at a : it is the limit lim x ! a x<a f ( x ) f ( a ) x a = f 0 ( a ) If f is di§erentiable (resp. right-di§erentiable ; resp. left-di§erentiable) at a; then it is continuous as well (resp. right-continuous, resp. left-continuous). Example of a function continuous everywhere but not di§erentiable at 0 : f ( x ) = (4 p + 1) j x j 2 if 1 2 p +1 j x j 1 2 p = (4 p + 3) j x j + 2 if 1 2 p +2 j x j 1 2 p +1 Check lim sup x ! 0 x 6 =0 f ( x ) x = 1; lim inf x ! 0 x 6 =0 f ( x ) x = 1 Exercise 30 Example of a function continuous over the interval [0,1], but not di§ erentiable anywhere on [0,1[: x = 0 ; a 1 a 2 :::a n ::: decimal development f ( x ) = 0 ; 0 a 2 0 a 4 0 a 6 0 ::: Operations on di§erentiable functions: The following operations respects the property ì f is di§erentiable at a " : the sum of functions, their product, their quotient, their composition. The same applies to right-di§erentiability, or left-di§erentiability, except for the composition. Exercise 31 What is a correct statement for the composition of right- or left- di§ erentiable functions? Computation rules ( P i i f i ) 0 = P i i f 0 i where i ; i = 1 ; :::; n; are constant ( f 1 f 2 ::: f n ) 0 = P i f 0 i ( f 1 ::: f i 1 f i +1 ::: f n ) ( 1 f ) 0 = f 0 f 2 ( f g ) 0 ( x ) = f 0 ( g ( x )) g 0 ( x ) chain rule If f is di§erentiable at x and f 0 ( x ) 6 = 0 ; then f is a bijection from an interval around x (i.e., an interval containing x as an interior point), to an interval around f ( x ) ; and the inverse function f 1 is di§erentiable at y = f ( x ) ; with the following derivative: ( f 1 ) 0 ( y ) = 1 f 0 ( f 1 ( y )) = 1 f 0 ( x ) Successive derivatives are denoted f 0 ; f " ; f (3) ; :::; f ( p ) ; ::: f is in the class C p ; or simply ì f is C p " on an open interval A if f ( p ) exists everywhere in A and is continuous on A: If f ( p ) exists at a; we say that f is p times di§erentiable at a ; in this case f is C p 1 in an interval around a; but not necessarily C p : 13 Exercise 32 Example of a function di§ erentiable everywhere on R but of which the derivative is not a continuous function: f ( x ) = x 2 sin 1 x if x 6 = 0 f (0) = 0 The set of C p functions is stable (respected) by the sum, the product, the quotient and the composition of functions. Rolle theorem : If f is continuous on the interval [ a; b ] in R ; and di§erentiable on ] a; b [ , there exists a number c 2 ] a; b [ such that f ( b ) f ( a ) = ( b a ) f 0 ( c ) This fundamental result has many applications. Fix any interval I of R and a function f continuous on I and di§erentiable in all interior points of I : 1. f is constant on I if and only if f 0 is the null function, 2. f is non decreasing if and only if f 0 is non negative on the interior of I; 3. f is (strictly) increasing if and only if f 0 is positive on the interior of I: Local extremum: we say that x is a local extremum of f if there is an interval V I around x such that f reaches at x its maximum on V or its minimum on V (we speak of a local maximum, or a local minimum). Under the same premises as above: 4. x is a local extremum of f only if f 0 ( x ) = 0 : Let f; g be two functions continuous on [ a; b ] and di§erentiable on ] a; b [: 5. fj f 0 ( x ) j g 0 ( x ) for all x 2 ] a; b [ g ) j f ( b ) f ( a ) j g ( b ) g ( a ) ; in particular 6. fj f 0 ( x ) j K for all x 2 ] a; b [ g ) j f ( b ) f ( a ) j K ( b a ) Taylor development: This is a more sophisticated application of Rolleí s theorem, allowing to ap- proximate a C n +1 function by a polynomial of degree n: Global Taylor formula: Let f be of class C n on the (bounded) interval [ a; b ] and di§erentiable ( n +1) times on ] a; b [; then there exists c 2 ] a; b [ such that: f ( b ) = f ( a ) + n P k =1 ( b a ) k k ! f ( k ) ( a ) + ( b a ) n +1 ( n +1)! f ( n +1) ( c ) Corollary: if the ( n + 1) th derivative satisÖes f ( n +1) ( c ) K for all c 2 ] a; b [ ; then we have the following approximation: f ( b ) f ( a ) n P k =1 ( b a ) k k ! f ( k ) ( a ) K ( b a ) n +1 ( n +1)! Local Taylor formula: Let f be n times di§erentiable at a; then the function ( x ) deÖned by: 14 (0) = 0 and for all x 6 = 0 : f ( x ) = f ( a ) + n P k =1 ( x a ) k k ! f ( k ) ( a ) + ( x a ) n ( x ) satisÖes lim x ! 0 ( x ) = 0 : Applications: 1. A polynomial of degree n coincides with its Taylor development of order n: 2. If f and g have the same k th derivative at a; for k = 0 ; 1 ; 2 ; :::; n; the di§erence ( f g )( x ) = ( x a ) n ( x ) is a very áat function. 3. The exponential function is C 1 and coincides with its ìTaylor seriesî e x = 1 + 1 P n =1 x n n ! : Example of a C 1 function that does not coincide with its ìTaylor seriesî : f ( x ) = e 1 x 2 for x 6 = 0; f (0) = 0 : Note that f is C 1 and f ( k ) (0) = 0 for any k = 0 ; 1 ; 2 ::: But f is á atter than any polynomial at 0 : 15 Chapter 3: Linear Algebra 1. The vector space R n The elements of R n are called vectors. A linear combination of the vectors x 1 ; :::; x p takes the form 1 x 1 + ::: + p x p where 1 ; :::; p are real numbers. The set of vectors f x 1 ; :::; x p g is linearly independent if for all 1 ; :::; p ; f 1 x 1 + ::: + p x p = 0 g ) f 1 = ::: = p = 0 g : Exercise 33 This implies x i 6 = x j if i 6 = j; and x i 6 = 0 all i (but the converse implication is false). Example: f x 1 ; x 2 g is linearly independent i§ x 1 ; x 2 are not collinear. A vector space (or subspace, or linear subspace) is any subset X R n stable by addition of vectors and scalar multiplications: for all x; y 2 R n f x; y 2 X g ) f x + y 2 X g for all x 2 R n ; all 2 R f x 2 X g ) f x 2 X g Equivalently, X is a vector space i§ it contains all linear combinations of its elements. If S is any subset of R n ; the set E ( S ) of all linear combinations of vectors in S is a vector space, called the span of S : E ( S ) = f x 2 R n 9 p 2 N ; 9 x 1 ; :::; x p 2 S; 9 1 ; :::; p 2 R : x = 1 x 1 + ::: p x p Examples: The span of a single (non zero) vector is a straight line (containing 0 ), the span of two linearly independent vectors is a plane. Dimension of a vector space. dim X = r reads: the dimension of the vector space X is r ( r is a nonnegative integer), and means: X contains a linearly independent set of r vectors but none of r + 1 : Conventionally the null space {0} has dimension 0. Examples: straight lines and planes have respective dimensions 1 and 2 : Basis of a vector space A basis of the vector space X is a linearly inde- pendent subset of X spanning X: Equivalently, it is a subset of dim X linearly independent vectors of X: If f x 1 ; :::; x r g is a basis of X; each vector x in X is uniquely written as a linear combination of the basis: x = 1 x 1 + ::: + r x r = r P i =1 i x i ; for a unique set of numbers 1 ; :::; r called the coordinates of x: Example: The standard basis f e 1 ; :::; e n g of R n : e i has all its coordinates 0 except 1 at the i th coordinates. Important fact: If the vector space X is spanned by a set of r vectors, then dim X r: Corollary: dim R n = n: The incomplete basis theorem : If f x 1 ; :::; x r g is a linearly independent subset of the vector space X (hence r r = dim X ) we can Önd vectors x r +1 ; :::; x r in X such that f x 1 ; :::; x r g is a basis of x: In particular X has a basis. The sum of the subsets S; T R n is deÖned as: 16 S + T = f x 2 R n 9 y 2 S; z 2 T : x = y + z g Remark: S + S 6 = 2 S: If X is a vector space, we have X + X = X: A sum of vector spaces is a vector space. Direct sum of subspaces: If X 1 ; X 2 are two vector spaces such that X 1 \ X 2 = f 0 g we say that the sum X 1 + X 2 is direct. In general the sum of vector spaces X 1 + ::: + X p = Y is direct if and only if (deÖnition) we have for all x 1 ; :::; x p f x i 2 X i for all i = 1 ; :::; p and p P i =1 x i = 0 g ) f x i = 0 for all i = 1 ; :::; p g In this case every vector y 2 Y has a unique decomposition as y = p P i =1 x i where x i 2 X i for all i = 1 ; :::; p Exercise 34 If the sums X 1 + X 2 and X 2 + X 3 are direct, the sum X 1 + X 2 + X 3 is not necessarily direct. Why? Example: if each X i is a straight line spanned by x i ; the sum X 1 + ::: + X p is direct i§ the set f x 1 ; :::; x p ) is linearly independent. This generalizes: The sum X 1 + ::: + X p is direct i§ when we pick a basis f x i 1 ; :::; x i r i g for X i ; for i = 1 ; :::; p; the set f x 1 1 ; :::; x 1 r 1 ; x 2 1 ; :::; x p 1 ; :::; x p r p g is linearly independent. 2. The Euclidean Space R n The scalar product of two vectors x; y in R n is deÖned by: x y = n P i =1 x i y i ; where x = ( x 1 ; :::; x n ) ; y = ( y 1 ; :::; y n ) The scalar product is commutative, and linear in each variable: for all x; x 0 ; y 2 R n ; all ; 2 R : x y = y x; ( x + x 0 ) y = x y + x 0 y The euclidean norm of a vector x in R n is deÖned by: j x j = ( n P i =1 x 2 i ) 1 2 = ( x x ) 1 2 Properties of the norm: for all x; y 2 R n ; all 2 R : i) j x j 0; j x j = 0 if and only if x = 0 ii) j x j = j j j x j iii) j x y j j x j j y j (Schwartzí s inequality) iv) j x + y j j x j + j y j (triangular inequality) v) In (iii) (resp. iv) equality holds if and only if f x; y g are linearly dependent. Say that x and y are orthogonal if x y = 0 : For any x; y; we have j x + y j 2 = j x j 2 + j y j 2 + 2 x y; hence j x + y j 2 = j x j 2 + j y j 2 if and only if x y = 0 (Pythagoraí s formula) j x y j 2 + j x + y j 2 = 2( j x j 2 + j y j 2 ) We say that f x 1 ; :::; x p g is an orthogonal set of vectors if x i ; x j are pairwise orthogonal for all i; j: 17 An orthogonal set of vectors where each vector is nonzero is linearly inde- pendent. An orthonormal set of vectors is an orthogomal set of vectors f x 1 ; :::; x p g such that, in addition, j x i j = 1 for all i = 1 ; :::; p: In particular, it is linearly independent. An important fact: Every vector space of R n has an orthonormal basis. For instance, the standard basis in an orthonormal basis of R n : In an orthonormal basis f x 1 ; :::; x p g of a subspace, the coordinates of y are y = p P i =1 ( y x i ) x i : The incomplete basis theorem extends to orthonormal basis: if f x 1 ; :::; x r g is an orthonormal set of vectors in the vector space X; we can Önd vectors f x r +1 ; :::; x r g in X such that f x 1 ; :::; x r g is an orthonormal basis of X: Corollary: If X is a vector space of R n with dimension r ; r < n; we can Önd an orthonormal set of n r = p vectors z 1 ; :::; z p in R n such that: x 2 X if and only if x z 1 = ::: = x z p = 0 In this way, X is represented by p linear equations. Example: A vector space with dimension n 1 is called a hyperplane. It is described by a single equation X = f x 2 R n =x z = 0 g ; for some nonzero vector z 2 R n : 3. Linear operators, matrices The mapping A from R n to R m (where n and m are arbitrary integers) is linear if for all x; y 2 R n ; all 2 R : A ( x + y ) = Ax + Ay A ( x ) = Ax This implies that A commutes with linear combinations: A ( n P i =1 i x i ) = n P i =1 i Ax i The set of linear operators from R n into R m is denoted L ( R n , R m ) : It is a vector space for the addition of operators and multiplication by a real number. In the standard basis of R n and R m ; a linear operator A is written as a m n matrix A = [ a ij ] where i is the index of rows, i = 1 ; :::; m; and j is that of columns, j = 1 ; :::; n: The jth column of A is formed by the coordinates of Ae j so that: A ( n P i =1 j e j ) = n P j =1 j Ae j = ( n P j =1 a 1 j j ; n P j =1 a 2 j j ; :::; n P j =1 a m j ) The composition of linear operators respects linearity. It corresponds to the multiplication of matrices. A 2 L ( n; m ) ; B 2 L ( m; p ) ) C = B A 2 L ( n; p ) or in matrix form A = [ a ij ] ; B = [ b ki ] ) BA = [ c kj ] c kj = n P i =1 b ki a ij k = 1 ; :::; p ; i = 1 ; :::; m ; j = 1 ; :::; n 18 Matrix notation for linear operators: we identify x 2 R n with the operator in L (1 ; n ) : t ! t x; of which the matrix is the column vector formed by the coordinates of x: Then the equation y = Ax is simply a product of matrices, ( m; n ) ( n; 1) = ( m; 1) : For instance the matrix of a linear form y 2 L ( R n ; R ) is a row vector and y x = P i y i x i is both the scalar product of y and x (vectors in R n ) and the value of the linear form y at x: Computational properties the multiplication is associative (AB)C=A(BC) and distributive w.r.t. addition and scalar multiplication: ( A + A 0 ) B = AB + A 0 B A ( B + B 0 ) = AB + AB 0 Caution: the multiplication of square matrices (i.e., matrices of dimension n n ) is not commutative. Exercise: give an example with square matrices of size 2. Norm of a matrix/linear operator: If j x j stands for the euclidean norm in R n or R m ; the norm of A 2 L ( n; m ) is deÖned as: k A k = sup j Ax j where x is any vector such that j x j 1 : The norm k A k is positive if A is not the zero operator ( Ax = 0 for all x ) ; and it is bounded above as follows: A = [ a ij ] ) k A k n: sup i;j j a ij j The norm meets all the properties listed in Section 4.1. below. The kerne l of A 2 L ( n; m ) is the inverse image of 0 : x 2 ker( A ) , Ax = 0 : The range of A is just A R n (same deÖnition as for any mapping). Both the kernel and range are vector spaces, respectively of R n and R m : Important fact for all A 2 L ( n; m ) : dim(ker A ) + dim( range A ) = n: The dimension of range ( A ) is called its rank Fix a linear operator A 2 L ( n; m ) : Then: i) A has full rank (rank A = m ) only if n m ii) A is one-to-one ( Ax = Ay ) x = y ) if and only if its kernel is f 0 g which is equivalent to rank ( A ) = n: This is possible only if n m: iii) A is a bijection if and only if f n = m and A has full rank g or equivalently f n = m and A is one-to-one}. In the latter case we say that A is invertible . We denote by I ( n ) the set of invertible operators from R n into itself. A noninvertible element of L ( n; n ) is called singular. Thus an invertible operator is also called nonsingular. The set I ( n ) is open in the space L ( n; n ) : If A 2 I ( n ) and k B k < 1 k A 1 k ; then A + B 2 I ( n ) : Thus we can add a small matrix to an invertible operator and still avoid singular operators. 4. The Algebra of Square Matrices 19 A linear operator from R n into itself is written as a square matrix n n: Their set is denoted M ( n ) : Endowed with addition, scalar multiplication, and multiplication it is an algebra >From the above discussion we get: A 2 I ( n ) i§ for some B 2 M ( n ) : AB = BA = I In this case the matrix B is unique and denoted B = A 1 : A matrix A 2 M ( n ) is invertible i§ it is onto ( A R n = R n ) , and i§ it is one-to-one ( Ax = 0 ) x = 0) : Computation rules on inverse matrices: A 2 I ( n ) ) A 1 2 I ( n ) AB 2 I ( n ) ) A; B 2 I ( n ) ( :A ) 1 = 1 A 1 when 2 R f 0 g ; ( A 1 ) 1 = A ; ( AB ) 1 = B 1 A 1 Determinants. The transposed of the m n matrix A = [ a ij ] is the n m matrix t A = [ a ji ] : We can write any n n matrix as the list of its column vectors A = [ a 1 ; ::::a n ] ; all in R n : Thus t A = [ b 1 ; ::; b n ] where b i is the i th row vector of A: Denote S n = A ( f 1 ; :::; n g ) the set of permutations of f 1 ; :::; n g and recall from Chapter 1