Real Analysis with Economic Applications - Chapter J doc

... and continuity in conjunction, along with a s atis fac tory geom etric analysis. When the norm under consideration is apparent from the context, it is customary to dispense with the notation (X, ... Y ? (Yes!) 480 Chapter J Norm ed Linear Spaces This chapter introduces a v ery im portan t subclass of metric linear spaces, namely, the class of normed linear space s. We begin with...

Ngày tải lên: 04/07/2014, 10:20

... subset of R n with nonempty interior. Where e j denotes the jth unit vec tor in R n ,thejth partial deriva tive of ϕ ∈ R S at x ∈int R 2 (S) is defined as ∂ j ϕ(x) := lim ε→0    ϕ(x+εe j )−ϕ(x) ε    ,j= 1, ... derivativ e s of each Φ i ∈ R O at x exist, and we have D Φ,x (t 1 , , t n )=  n j= 1 ∂ j Φ 1 (x)t j , , n j= 1 ∂ j Φ m (x)t j  for all (t 1 , , t n ) ∈ R n ,...

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... we sa y that f is an injection (or a one-to-one, or an injective function/m ap). Finally, if f is bo th injective and surject ive, the n it is called a bijection (or a bije ctive function/map). ... x k−1 x and x −k := (x k ) −1 . For any integers i and j, provethatx i x j = x i +j and (x i ) j = x ij for any x ∈ X, and x i /x j = x i j and (y/x) i = y i /x i for any x ∈ X\{0}. Wh ile...

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... to be a well-ordering if ev ery nonempty subset of X has a -minimum. In this case X is said to be w e ll-ordered by ,and(X, ) is called a well-ord e red s et,orshortlyawoset. Well-ordered sets ... theory), ev en t hough the former see ms impossible, and the latter self-evident. 75 x  b j  a j  y so that ϕ j (x)=1> 0=ϕ j (y). Since x  y implies ϕ i (x) ≥ ϕ i (y) for every i,...

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... matrix A := [a ij ] n×n (Section A.1.6) and any x ∈ R n , we write Ax for the n-vector (  n a 1j x j , ,  n a nj x j ). Give an example of an n × n matrix A such that the map x → Ax is a con ... R ∞ since the mem bers of such a space are real sequences that are either bounde d or that satisfy some form of a summability condition (that ensures that d p is real- valued). Indeed, no...

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... semicon tinuity of a function like ϕ • whic h ma y be extended real- valued (ev en w hen ϕ is real- valued). 20 Exercise 32. Let ϕ be any real function defined on a metric space X. Pro ve: (a) ϕ • ≤ ... prove this approximation-by-continuous-functions theorem, but you might want to try it out for yourself in the case where X is a compact interval. 176 3.2 The Local-to-Global Method Since a...

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... strategic game G := {(X i , π i ) i=1, ,m }, we let X −i := {(ω 1 , ,ω m−1 ):ω j ∈ X j for j& lt;iand ω j 1 ∈ X j for j& gt;i} 260 Intuitively speaking, upper hemicontinuity at x sa ys that a small ... where π i (x)=H i  x i , m  j= 1 x j  with H i : X i ×{  m x j : x ∈ X} → R being an arbitrary function, i =1, , m. This sort of a game i s called an aggregative game. 28 As...

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... αL(x).) E{dpsoh 6. For any m, n ∈ N, A ∈ R m×n and x ∈ R n , we define Ax := # n S j= 1 a 1j x j , , n S j= 1 a mj x j $ , where A := [a ij ] m×n (Section A.1.6.). Then L : R n → R m defined by L(x):=Ax is a linear ... have L i (x)=L i # n S j= 1 x j e j $ = n S j= 1 L i (e j )x j for each i =1, , m. (Here {e 1 , , e n } is the stan da r d basis for R n (Exa m ple 5.[2]).) The...

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... that S n y i v i =0and S n y i u j i =0for all j =1, , m. Exercise 60. H Recall that an n × n matrix [a ij ] n×n is said to be sto chastic if a ij ≥ 0 for a ll i an d j, and S n j= 1 a ij =1for all i. Prove ... al-int X (S), then B\{y, z } ⊆ al-int X (S). (Why?) But this wou ld imply that x ∈ al-int X (S), a contradiction. Th us al-int X (S)∩B = ∅. In turn, this means that al- int X (A...

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... itrary a j ∈ arg max{E q σ (· ,j) (u(·, a(·))) : a ∈A}. Then V (J, q) σ (u)= S j J S ω∈Ω σ(ω)  S i∈I p(ω,i)Θ(i, j)  u(ω, a j (ω)) = S j J S i∈I Θ(i, j) S ω∈Ω σ(ω)p(ω,i)u(ω, a j (ω)) ≤ S i∈I S j J Θ(i, ... than (J, q), th at is, there exists a map Θ ∈ R I J + suc h that S j J Θ(· ,j) =1, and q(ω ,j) = S i∈I p(ω,i)Θ(i, j) for each (ω ,j) ∈ Ω × J. Take any (σ,u) ∈ P(Ω) ×...

Ngày tải lên: 04/07/2014, 10:20