Mathematics for Economists

Transcription

1 Mathematics for Economists Victor Filipe Sao Paulo School of Economics FGV Metric Spaces: Basic Definitions Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

2 Definitions and Examples Let X be a non-empty set Definition A function d : X X R + that satisfies the following properties is called a distance function (or a metric) on X: For any x, y, z X, (i) d(x, y) = 0 if, and only if, x = y; (ii) (Symmetry) d(x, y) = d(y, x); (iii) (Triangle Inequality) d(x, y) d(x, z) + d(z, y). The pair (X, d) is called a metric space If d satisfies (ii) and (iii) and d(x, x) = 0 for any x X, then we say that d is a semi-metric on X Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

3 Definitions and Examples Discrete metric d(x, y) := { 1 if x y 0 if x = y Standard metrics on R n ( n ) 1/p d p (x, y) := x i y i p for 1 p < i=1 and d (x, y) := max{ x i y i : i = 1,..., n} The space (R n, d 2 ) is called the n-dimensional Euclidean space Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

4 Definitions and Examples Proposition (Minkowski s Inequality 1) For any n N, a i, b i R, i = 1,..., n and any 1 p <, ( n ) 1/p ( n ) 1/p ( n ) 1/p a i + b i p a i p + b i p i=1 i=1 i=1 We denote by C p the unit circle for the metric d p where 0 = (0, 0) C p := {x R 2 : d p (0, x) = 1} Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

5 Definitions and Examples Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

6 Definitions and Examples Standard metric on R d (x, y) := f(x) f(y) where f : R [ 1, 1] is the bijection defined by f( ) := 1, f( ) := 1, and f(t) := t 1 + t l space and the sup-metric l := {(x n ) R N : sup{ x n : n N} < } and let d ((x n ), (y n )) := sup{ x n y n : n N} Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

7 Definitions and Examples l p space For any 1 p < we let l p := {(x n ) R N : x i p < } i=1 and let ( ) 1/p d p ((x n ), (y n )) := x i y i p i=1 Proposition (Minkowski s Inequality 2) For any (a n ), (b n ) R N and any 1 p <, ( ) 1/p ( ) 1/p ( ) 1/p a i + b i p a i p + b i p i=1 i=1 i=1 Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

8 Proof of Minkowski s Inequality Let α := ( i=1 a i p ) 1/p and β := ( i=1 b i p ) 1/p Pose â i := (1/α) a i and ˆb i := (1/β) b i Show that ( α a i + b i p (α + β) p α + β â i + β ) p α + β ŷ i Use the convexity of t t p to show that ( α a i + b i p (α + β) p α + β â i p + β ) α + β ŷ i p Deduce that a i + b i p (α + β) p i=1 Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

9 Definitions and Examples B(T ) and the sup-metric The space of bounded functions B(T ) := {f R T : sup{ f(x) : x T } < } and the sup-metric d (f, g) := sup{ f(x) g(x) : x T } Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

10 Definitions and Examples Consider a metric space (X, d) We can endow any subset Y X with the metric d Y Y We abuse notation and write (Y, d) The space C[a, b] can be endowed with the metric d since C[a, b] B[a, b] The space C 1 [a, b] can be endowed with the metric d but also with the metric d, (f, g) := d (f, g) + d (f, g ) Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

11 Open and Closed Sets We fix (X, d) a metric (or semi-metric) space Definition For any x X and ε > 0, we define the ε-neighborhood of x in X as the set N ε,x (x) := {y X : d(x, y) < ε} The set N ε,x (x) is also denote B X (x, ε) as the open ball (in X) centered at x with d-radius ε Definition A subset Y X is said to be a neighborhood of x in X if it contains an ε-neighborhood of x in X for at least one ε > 0, i.e., ε > 0, B X (x, ε) Y Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

12 Open and Closed Sets We endow the space R 2 (and any subset) with the Euclidean metric d 2 The 1-neighborhood of 0 = (0, 0) in R 2 is {(x 1, x 2 ) R 2 : x x 2 2 < 1} The 1-neighborhood of 0 = (0, 0) in R {0} is {(x 1, 0) R 2 : 1 < x 1 < 1} Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

13 Open and Closed Sets Definition A subset S of X is said to be open in X if x S, ε > 0, B X (x, ε) S A subset S is open in X if, and only if, it is a neighborhood of each of its points Definition A subset S of X is said to be closed in X if X \ S is open in X A subset may be neither closed nor open Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

14 Open and Closed Sets Let X be a metric space (with metric d) and S X Definition The largest (for inclusion) open set in X that is contained in S is called the interior of S relative to X and is denote by int X (S) (or int(s)) Definition The smallest (for inclusion) closed set in X that contains S is called the closure of S relative to X and is denote by cl X (S) (or cl(s)) Definition The boundary of S relative to X, denoted by bd X (S) (or bd(s)) is defined as bd X (S) := cl X (S) \ int X (S) Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

15 Open and Closed Sets Let S Y X One may define the interior of S relative to X or Y We may well have int X (S) int Y (S) Take X = R R, Y = R {0} and S = [0, 1] {0} We have int X (S) = and int Y (S) = (0, 1) {0} Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

16 Open and Closed Sets The open ball B X (x, ε) = N ε,x (x) is open Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

17 Open and Closed Sets We denote by B X (x, ε) the set B X (x, ε) := {y X : d(x, y) ε} The set B X (x, ε) is closed, it is called the closed ball centered at x with radius ε We have B X (x, ε) int [ ] B X (x, ε) Do we always have the converse containments? and cl [B X (x, ε)] B X (x, ε) Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

18 Open and Closed Sets Fix a metric space (X, d) The sets X and are open and closed The singleton {x} is closed If d is the discrete metric then any subset is open and closed Denote by O X the class of all open subsets of X Proposition The union of any collection of open sets is open, i.e., if C O X then [ ] {S : S C} = S O We then get that the intersection of any collection of closed sets is closed The intersection of finitely many open sets is open, and the union of finitely many closed sets is closed Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34 S C

19 Open and Closed Sets The interior int(s) is the maximum element 1 of the set {V O X : V S} The maximum always exists since we have 2 Similarly, we have int(s) = {V O X : V S} cl(s) = {C C X : S C} where C X is the class of closed sets in X 1 For the binary relation. 2 Recall that O X. Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

20 Open and Closed Sets Exercises 1 S is open in X if, and only if, int X (S) = S 2 S is closed in X if, and only if, cl X (S) = S 3 x int X (S) if, and only if, ε > 0, B X (x, ε) S 4 x cl X (S) if, and only if, ε > 0, B X (x, ε) S 5 x bd X (S) if, and only if, ε > 0, B X (x, ε) S and B(x, ε) X \ S Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

21 Open and Closed Sets Exercise Let S be a closed subset of a metric space X, and x X \ S. There exists an open subset V of X such that S V and x X \ V Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

22 Convergent Sequences Let X be a metric space, x X and (x n ) X N Definition The sequence (x n ) converges to x if lim d(x n, x) = 0. We write x n x or lim x n = x d lim x n = x if, and only if, ε > 0, N N, n N, x n B X (x, ε) A sequence can converge to at most one limit Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

23 Convergent Sequences A sequence (x n ) is convergent in a discrete space (space endowed with the discrete metric) if, and only if, it is eventually constant, i.e., N N, n N, x n = x N Consider the space R k for some k N endowed with the metric d p for some p [1, ). A sequence (x n = (x n 1,..., xn k )) is convergent to x = (x 1,..., x k ) if, and only if, i = 1,..., k lim n xn i = x i. This property is independent of the choice of the metric d p Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

24 Convergent Sequences Let X := R N and consider the following real sequences and x n := ( 0,..., 0, 1 }{{} n, 0,...), yn := ( 0,..., 0, 1, 0,...) }{{} (n 1) times (n 1) times z n := ( 1 n,..., 1, 0,...) }{{ n} n times lim x n = 0 in l p but (y n ) is not convergent in l p lim z n = 0 in l but (z n ) does not converge to 0 in l 1 Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

25 Convergent Sequences Let f n B[0, 1] be defined by f n (t) := t n We have t [0, 1], lim f n(t) = 1 {1} (t) n However, the sequence (f n ) does not converge to 1 {1} for the sup-metric d Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

26 Sequential Characterization of Closed Sets Proposition A set S in a metric space X is closed if, and only if, every sequence in S that converges in X has its limit point in S S is closed if, and only if, (x n ) S N, x X, lim x n = x = x S Let X = (0, 1). Is the set (0, 1) closed in X for the standard (Euclidean metric)? Observe that (x n ) n N := (1, 1 2, 1,...) 0 3 Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

27 Sequential Characterization of Closed Sets Proposition For any subset S of a metric space X, the following statements are equivalent (i) x cl X (S) (ii) Every open neighborhood of x in X intersects S (iii) There exists a sequence in S that converges to x Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

28 Equivalence of Metrics If d is a metric on X, we denote by O(d) the class of all open subsets of X with respect to d. Definition Two metrics d and d on the same space X are said to be equivalent (in X) if the respective classes of open sets coincide, i.e., O(d) = O(d ) Lemma If d and d are equivalent then a sequence converges with respect to d if, and only if, it converges with respect to d (to the same limit) Example The metrics d and d/(1 + d) are equivalent Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

29 Equivalence of Metrics Definition A subset S of a metric space X is said to be bounded if there exists M > 0 and x S such that S B X (x, M) If S is not bounded, then it is said to be unbounded A finite union of bounded sets is bounded If (X, d) is a metric space, then any set S X is bounded with respect to the metric d/(1 + d) Boundedness is not invariant under equivalence of metrics Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

30 Equivalence of Metrics Definition Two metrics d and d on the same space X are said to be strongly equivalent (in X) if there exists α, β > 0 such that αd d βd If d and d are strongly equivalent then they are equivalent Boundedness is invariant under strong equivalence of metrics The metrics d 1 and d 1 are equivalent in [0, 1] but not strongly equivalent In R n, any metric d p with 1 p < is strongly equivalent to d since d d p nd In R n, the metrics d p and d q are strongly equivalent for any 1 p, q Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

31 Finite Products Fix k N and consider metric spaces (X i, d i ) for each i = 1,..., k We denote by X := k i=1 X i the product space where a vector in X is denoted by x = (x 1,..., x k ) We can endow X by several metrics ρ 1 (x, y) := k d i (x i, y i ), i=1 ρ (x, y) := max i=1,...,k d i(x i, y i ) and, for each p 1 ( k ) 1/p ρ p (x, y) := (d i (x i, y i )) p i=1 Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

32 Finite Products All the metrics, ρ 1, ρ p and ρ are equivalent We choose the metric ρ := ρ 1 and refer to it as the product metric Proposition A sequence (x n ) in X converges to x for the product metric if, and only if, lim n x n i = x i for each i = 1,..., k Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

33 Countably Infinite Products Let (X i, d i ) be a metric space for each i N The infinite product is denoted by X := i N X i We let ρ : X X R + be defined by ρ(x, y) := i=1 1 2 i min{1, d i(x i, y i )} This is called the product metric Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34

34 Countably Infinite Products Proposition The metric space (X, ρ) is bounded Proposition Let (x n ) be a sequence in X, i.e., x n = (x n i ) i I. We have lim x n = x i N, lim x n n i = x i Fix a, b R with a < b The set [a, b] N of all real sequences (x n ) with x n [a, b] is called the Hilbert Cube Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb / 34