Lecture 3. Econ August 12

Transcription

1 Lecture 3 Econ August 12

2 Lecture 3 Outline 1 Metric and Metric Spaces 2 Norm and Normed Spaces 3 Sequences and Subsequences 4 Convergence 5 Monotone and Bounded Sequences Announcements: - Friday s exam will be at 3pm, in WWPH 4716; recitation will be at 1pm. The exam will last an hour.

3 Vector Space Over the Reals Definition A vector space V over R is a 4-tuple (V, R, +, ) where V is a set, + : V V V is vector addition, : R V V is scalar multiplication, satisfying: 1 Closure under addition: x, y V, (x + y) V 2 Associativity of +: x, y, z V, (x + y) + z = x + (y + z) 3 Commutativity of +: x, y V, x + y = y + x 4 Existence of vector additive identity:!0 V s.t. x V, x + 0 = 0 + x = x 5 Existence of vector additive inverse: x V!y V s.t. x + y = y + x = 0. Denote this unique y as ( x), and define x y as x + ( y). 6 Distributivity of scalar multiplication over vector addition: α R x, y V, α (x + y) = α x + α y 7 Distributivity of scalar multiplication over scalar addition: α, β R x V, (α + β) x = α x + β x 8 Associativity of : α, β R, x V, (α β) x = α (β x) 9 Multiplicative identity: x V, 1 x = x

4 Fun With Vector Spaces Supose we have a vector space V over R and x V. What is 0 x =? distributivity of scalar muliplication existence of vector additive identity {}}{{}}{ 0 x = (0+0) x = 0 x+0 x = 0 where the first zero is a scalar while the last is a vector. Using this result: 0 = 0 x = (1 1) x = 1 x + ( 1) x = x + ( 1) x So we conclude ( 1) x = ( x). which is not something implied by the nine properties of a vector space.

5 Distance We work with abstract sets (typically vector spaces) and our main objective is to be able to talk about their properties as well as properties of functions defined over them. For example, we would like to define notions like limits and convergence, continuity of a function from one of these sets to another, and so on. Continuity of a function f : R R is defined as ε > 0 δ > 0 such that x y < δ f (x) f (y) < ε How can we construct a similar definition when the domain is some abstract set X? ε, δ, and f ( ) are scalars, but we do not know how to measure the distance between two points. So, next we define function that measures distance between any two points in a set. using this function, one can express the idea that two points are close (say within δ of each other). Notice that the even more general case in which f is a function from X to Y does not complicate things. First, we define distance for abstract settings; next, specialize to vector spaces.

6 Metric Spaces and Metrics Definition A metric d on a set X is a function d : X X R + satisfying 1 d(x, y) 0, with d(x, y) = 0 x = y x, y X. 2 d(x, y) = d(y, x) x, y X (symmetry). 3 triangle inequality: d(x, z) d(x, y) + d(y, z) x, y, z X A metric space (X, d) is a set X with a metric d defined on it. The value d(x, y) is interpreted as the distance between x and y. This definition generalizes the concept of distance to abstract spaces. Why triangle inequality? y x z Notice that a metric space need not be a vector space.

7 Metrics: Examples Example The discrete metric on any given set X is defined as { 0 if x = y d(x, y) = 1 if x y It trivially satisfies 1. and 2., and as for triangle inequality: 1 2. Example The l 1 metric (taxicab) on R 2 is defined as d : R 2 R 2 R + : d(x, y) = x 1 y 1 + x 2 y 2 Example The l metric on R 2 is defined as d : R 2 R 2 R + : d(x, y) = max { x 1 y 1, x 2 y 2 }

8 Metrics and Applications Definition In a metric space (X, d), a subset S X is bounded if x X and β R such that s S, d(s, x) β Given a metric space, boundedness is a propety of subsets (it is relative!). Definition In a metric space (X, d), we define the diameter of a subset S X by diam (S) = sup{d(s, s ) : s, s S} A metric can be used to define a ball of a given radius around a point. Definition In a metric space (X, d), we define an open ball of center x and radius ε as B ε (x) = {y X : d(y, x) < ε} and a closed ball of and center x and radius ε as B ε [x] = {y X : d(y, x) ε}

9 Distance Between Sets We can also define the distance from a point to a set: d(a, x) = inf d(a, x) a A and the distance between sets: d(a, B) = inf d(b, a) a A = inf{d(a, b) : a A, b B} Note that d(a, B) is not a metric (why?).

10 Norms and Normed Spaces The following defines a vector s length (its size). Definition Let V be a vector space. A norm on V is a function : V R + satisfying 1 x 0 x V. 2 x = 0 x = 0 x V. 3 triangle inequality: x + y x + y x, y V 4 αx = α x α R, x V Why is this also called triangle inequality? Definition x x y 0 x + y y x y A normed vector space is a vector space over R equipped with a norm. There are many different norms: if is a norm on V, so are 2 and 3 and k for any k > 0.

11 Normed Spaces Can one relate norm and distance on a given normed vector space? In R n, the standard norm is defined as ( n z = while the Euclidean distance between two vectors x and y d(x, y) = (x 1 y 1 ) 2 + (x 2 y 2 ) (x n y n ) 2 = n (x i y i ) 2 Norm and Euclidean distance Here, the distance between x and y is the length of the vector given by their difference: d(x, y) = x y i=1 z 2 i ) 1 2 i=1 This idea generalizes.

12 Norm and Distance In any normed vector space, the norm can be used to define a distance. Theorem Let (V, ) be a normed vector space. Let d : V V R + be defined by d(v, w) = v w Then (V, d) is a metric space. In a normed vector space, we define a metric using the norm. Remember that x : V R + is the lenght of a vector x V. Here x = v w. Hence, the distance between two vectors is the length of the difference vector.

13 Proof that (V, d) is a Metric Space I We must verify that the d defined satisfies the three properties of a metric. First property: d(x, y) 0, with d(x, y) = 0 x = y x, y X Proof. Let v, w V. By definition, d(v, w) = v w 0. Moreover d(v, w) = 0 v w = 0 v w = 0 by def of (v + ( w)) + w = w by vector additive inverse v + (( w) + w) = w by associative property v + 0 = w by vector additive inverse v = w

14 Proof that (V, d) is a Metric Space II For symmetry, we need to show that d(x, y) = d(y, x) x, y X Proof. Let v, w V. d(v, w) = v w = 1 v w = ( 1)(v + ( w)) by def of = ( 1)v + ( 1)( w) = v + w by the first slide = w + ( v) = w v = d(w, v)

15 Proof that (V, d) is a Metric Space III The triangle inequality is d(x, z) d(x, y) + d(y, z) x, y, z X. Proof. Let u, w, v V. d(u, w) = u w = u + ( v + v) w = u v + v w u v + v w by triangle for = d(u, v) + d(v, w) Summary Since we have shown that d( ) = satisfies the three properites of a metric on V, we conclude that (V, d) is a metric space.

16 Normed Vector Spaces: Examples R n admits many norms The n-dimensional Euclidean space R n supports the following norms the standard (Euclidean) norm: x 2 = n (x i ) 2 = x x where (the dot product of vectors) is defined as: n x y = x 1 y 1 + x 2 y x n y n = x i y i i=1 the L 1 norm: n x 1 = x i i=1 the L p norm: x = ( x 1 p x n p ) 1 p the maximum norm, or sup norm, or L norm: x = max{ x 1,..., x n } i=1

17 Normed Vector Spaces: Results Theorem (Cauchy-Schwarz Inequality) If v, w R n, then ( n ) 2 ( n v i w i i=1 i=1 v 2 i ) ( n i=1 w 2 i ) This can also be written as v w 2 v 2 w 2 or v w v w where is the standard norm. Prove this as an exercise.

18 Different Norms Different norms on a given vector space yield different geometric properties. In R n, a closed ball around the origin of radius ε is: B ε [0] = {x R n : d(x, 0) ε} = {0 X : x ε} Draw a ball in R 2 ({x R 2 : x = 1}) in 3 different cases: standard norm ( 2) L 1 norm ( 1) sup-norm ( ) x 2 = x x 2 2 = 1, x 1 = x 1 + x 2 = 1, x = max{ x 1, x 2 } = 1

19 Equivalent Norms Do different norm imply different results when applied to ideas like convergence, continuity and so on? Hopefully not. Definition Two norms and on the same vector space V are said to be Lipschitz-equivalent (or equivalent) if m, M > 0 such that x V, m x x M x Equivalently, m, M > 0 such that x V, x 0, m x x M

20 Equivalent Norms Equivalent norms define the same notions of convergence and continuity. For topological purposes, equivalent norms are indistinguishable. Compare Different Norms Let and be equivalent norms on the vector space V, and fix x V. Let Result B ε (x, ) = {y V : x y < ε} and B ε (x, ) = {y V : x y < ε} Then, for any ε > 0, B ε (x, ) B M ε(x, ) B ε (x, ) m ε balls are close to each other. In R n (or any finite-dimensional normed vector space), all norms are equivalent. Roughly, up to a difference in scaling, for topological purposes there is a unique norm in R n.

21 Sequences of Real Numbers Definition A sequence of real numbers is a function f : N R. Notation If f (n) = a n, for n N, we denote the sequence by the symbol {a n } n=1, or sometimes by {a 1, a 2, a 3,... }. The values of f, that is, the elements a n, are called the terms of the sequence. Examples Let a n = n then {a n } n=1 = {1, 2, 3,... } Let a 1 = 1, a n+1 = a n + 2, n > 1 then {a n } n=1 = {1, 3, 5,... } Let a n = ( 1) n, n 1 then {a n } n=1 = { 1, 1, 1, 1,... }

22 Limit of a Sequence of Real Numbers Definition A sequence of real numbers {a n } is said to converge to a point a R if ε > 0 there exists N(ε) N such that n N a n a < ε The sequence converges to a limit point if for every strictly positive epsilon, there exists a natural number (possibly dependent on epsilon) such that all the points in the sequence on and after that number are at most epsilon away from the limit point. Notation When {a n } converges to a we usually write or a n a lim a n = a n Generalize to arbitrary sequences next.

23 Sequences The objects along the sequence are now elements of some set. Definition Let X be a set. A sequence is a function f : N X. Notation We denote the sequence by the symbols {x n } n=1, or sometimes by {x n}. When X = R n each term in the sequence is a vector in R n.

24 Convergence To define convergence, X must be a metric space so that one talk about points being close to each other using the metric to measure distance. Definition Let (X, d) be a metric space. A sequence {x n } converges to x X if Notation ε > 0 N(ε) N such that n > N(ε) d(x n, x) < ε When {x n } converges to x we write x n x or lim n x n = x For any strictly positive epsilon, there exists a number such that all points in the sequence from then on are at most epsilon away from the limit point. Remark Similar to real numbers: we replaced the standard distance with a metric. The limit point must be an element of X.

25 Uniqueness of Limits for Real Numbers Theorem Let {a n } be a sequence of real numbers that converges to two real numbers b and b ; then, b = b. Proof. By contradiction. Assume (wlog) that b > b and let ε = 1 2 (b b ) > 0. By definition, there are N(ε) and N (ε) such that n N(ε) b a n < ε and n N (ε) b a n < ε But this is impossible since it implies that for n > max{n(ε), N (ε)}. 2ε = b b b a n + b a n < 2ε

26 Uniqueness of Limits Theorem (Uniqueness of Limits) In a metric space (X, d), if x n x and x n x, then x = x. Proof. By contradiction. Suppose {x n } is a sequence in X, x n x, x n x, and x x. Since x x, d(x, x ) > 0. Let ε = d (x,x ) 2 Then, there exist N(ε) and N n > N(ε) d(x (ε) such that n, x) < ε n > N (ε) d(x n, x ) < ε. Choose n > max{n(ε), N (ε)} Then a contradiction. d(x, x ) d(x, x n ) + d(x n, x ) < ε + ε = 2ε = d(x, x ) d(x, x ) < d(x, x )

27 Subsequences If {x n } is a sequence and n 1 < n 2 < n 3 < then {x nk } is called a subsequence. Definition A subsequence of the sequence described by f from N to X is the sequence given by a composite function f g : N X where and g(m) > g(n) whenever m > n. g : N N A subsequence is formed by taking some of the elements of the parent sequence in the same order. Example x n = 1 n, so {x n} = ( 1, 1 2, 1 3, 1 4, 1 5, 1 6,...). If n k = 2k, then {x nk } = ( 1 2, 1 4, 1 6,...).

28 Monotone Sequences of Reals Definitions A sequence of real numbers {x n } is increasing if x n+1 x n for all n. A sequence of real numbers {x n } is decreasing if x n+1 x n for all n. We say strictly increasing or decreasing if the above inequalities are strict. Definition A sequence of real numbers is monotone if it is either increasing or decreasing. Definition If {x n } is a sequence of real numbers, {x n } tends to infinity (written x n or lim n x n = ) if K R N(K) such that n > N(K) x n > K Similarly define x n or lim x n =.

29 Facts about Convergence of Sequences of Reals Convergence and Subsequences {a n } converges to b if and only if every subsequence of {a n } converges to b. Convergence Operations Let {a n } and {b n } be sequences such that lim n a n = a and lim n b n = b. Then 1 lim n (a n + b n ) = a + b 2 for c R, lim n ca n = ca, 3 lim n a nb n = ab 4 lim n (a n) k = a k 5 1 lim n a n = 1 a provided a n 0 (n = 1, 2, 3,... ), and a 0 6 a lim n n b n = a b provided b n 0 (n = 1, 2, 3,... ), and b 0

30 Boundedness for Sequences of Real Numbers Definition The sequence of real numbers {a n } is bounded above if m R such that a n m n N Definition The sequence of real numbers {a n } is bounded below if m R such that a n m n N

31 Boundedness and Convergence Theorem If a sequence of real numbers {a n } converges then it is bounded. Proof. Suppose a n a. Then, there is an integer N such that n > N implies a n a < 1. Define r as follows: r = max{1, a 1 a,..., a N a } Then a n a r for n = 1, 2, 3,....

32 Boundedness and Convergence Theorem Let {a n }, {b n }, and {c n } be sequences of real numbers such that and we have that both Then a n b n c n, n N a n a, c n a b n a Proof in the Problem Set.

33 Theorem Monotonicity and Convergence A monotone sequence of real numbers is convergent if and only if it is bounded. Proof. We have already shown that a convergent sequence is bounded (monotone or not), so we need only prove that a bounded and monotone sequence converges. Suppose {a n } is bounded and increasing. Let a = sup{a n : n N}. We want a to be the limit of {a n }; so, we need to show that ε > 0, N N such that a a n < ε, n N For any ε > 0, a ε is not an upper bound of {a n }; thus N such that a N > a ε. Since {a n } is increasing, for every n N we have a N a n Putting these observations together a ε < a N a n a where the last inequality follows from a n a, n. Thus, a n a < ε, n N Prove the case in which {a n } is bounded and decreasing as an exercise.

34 A Monotone Subsequence Always Exist Theorem (Rising Sun Lemma) Every sequence of real numbers has a monotone subsequence. Why the name? S U N Proof. Exercise

35 Bolzano-Weierstrass Theorem Theorem (Bolzano-Weierstrass) Every sequence of real numbers that is bounded has a convergent subsequence. Proof. This is done by using the Rising Sun Lemma and the previous theorem: every real sequence has a monotone subsequence and a monotone sequence is convergent if and only if it is bounded

36 Tomorrow We start defining open and closed sets, then we move to functions and define limits and continuity. 1 Open and Closed Set 2 Limits of functions 3 Continuity