Real Analysis with Economic Applications - Chapter K docx

... case we should work with Fréchet derivativ es. Well, that’s talking the “talk.” In this final section of the text, we aim to w alk the “walk” as well. As a case study, we tak e on the classical ... x m ) 2 ∞ 11 For t his to m ake sense I need to know that the two infinite series on the right-hand-side of ( 7 ) sum up to finite numbers. And I do know this. How? 513 w e seek t o be con tin uous...

Ngày tải lên: 04/07/2014, 10:20

... Theorem:Takeany(x kl ) ∈ R ∞×∞ such that S ∞ j=1 x kj converges for each k, and (x 1l ,x 2l , ) converges for each l. If there exists a real sequence (K 1 ,K 2 , ) such that |x kl | ≤ K l for each ... (Z\{0}) by (m, n) ∼ (k, l) iff ml = nk. The addition and multiplication operations on Q are then defined as [(m, n)] ∼ +[ (k, l)] ∼ =[(ml + nk, nl)] ∼ and [(m, n)] ∼ [ (k, l)] ∼ = [(mk, n...

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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 ... Principle, it contains a ⊇-maximal loset, say (Z i , ). By -densene ss of Y , Z i has neither a -maxim um nor a -minimum. Moreove r, by its maximalit y, it i s -d en s e in itself. By Cor .....

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... consider the set F K := {f α : α ∈ [0 ,K) } where K is a strictly positive extend ed real n u mber. It is easy to verify that F K is equicontinuous for any given K ∈ R ++ . To see this, pick an y x ∈ [0, ... I hav e not yet talked about the continuity of maps defined on an arbitrary Euclidean space. But I trust you know that all is kosher. If y o u have doubts, take a quick look at E...

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... uity of ϕ,soweget lim k ∞ lim m→∞ ϕ m (x k ) = lim k ∞ ϕ(x k )=ϕ(x) = lim m→∞ ϕ m (x)= lim m→∞ lim k ∞ ϕ m (x k ), that is, lim k ∞ lim m→∞ ϕ m (x k ) = lim m→∞ lim k ∞ ϕ m (x k ). (12) This is a ... ≤ |ψ k (x) − ψ k (z)| + |ψ k (z) −ψ l (z)| + |ψ l (z) −ψ l (x)| < ε for all k, l ≥ M. That is, |ψ k (x) − ψ l (x)| ≤ ε for any x ∈ T, or put differently, d ∞ (ψ k , ψ...

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... ϕ(x 0 ,x 1 ∗ )+  k i=1 δ i ϕ(x i ∗ ,x i+1 ∗ )+δ k+ 1 V (x k+ 1 ∗ ) for any k ∈ N. Bu t, thank s to (A1 ), V is bounded, so t here exists a K& gt;0 suc h that |V | ≤ K, and this clea rly e ntails th at δ k V (x k ) ... Therefore, for each x ∈ X, we have d(x, Φ(x)) = 1 k K (kd(x, Φ(x)) −Kd(x, Φ(x))) ≤ 1 k K (d(x, Γ(x)) −d(Φ(x), Γ(Φ(x)))) . Definin g ϕ ∈ R X by ϕ(x):= 1 k K d(x...

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... that the risk p refer- ences of the agent is state-independent. 305 for all A, B ⊆ N with A ∩ B = ∅. The capacit y v is said to be additive if v(A ∪ B)=v(A)+v(B) for all A, B ⊆ N with A ∩ B = ... = S n v({i})y i for all y ∈ R n + . (e)Letk ∈ N. Show that, for any (y,v i ,a i ) ∈ R n + ×V N × R + ,i=1, , k, we have ] yd  k S i=1 a i v i  = k S i=1 a i ] ydv i . (f ) (Gilboa-Schmeidl...

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... 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) = ∅ an d al-int X (A) ... of th e Hahn-Ban ach Extension Theo- rem 1 which also highlight s the importance of the Minkowsk i f un ction als for convex analysis. In the statem ent o f this result, and its many appl...

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... ou ld exist a k ∈ {1, ,n−1} such that a 1 , ,a k ≤ 0 and a k+ 1 , , a n ≥ 0 with S k a i < 0.Ifk = n − 1, this read ily con tra dicts (3), so conside r the case where n ≥ 3 and k ∈ {1, , n − ... u i is strictly increasin g , e k + S m x i ∗ ∈ S (where e k is the kth unit ve c tor in R l ,k= 1, , l). Therefore , the above inequality yields p k > 0 for each k. To comple...

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... θ m k x m k − θ m k y m k for e a ch k. Since t he m etric d of X is tra nslation invariant an d Y is a linear subspac e of X, therefore, d(θ m k x m k − θ m k y m k ,Y)=d(θ m k x m k ,Y + θ m k y m k )=d(θ m k x m k ,Y) ... subsequence (λ 1 (m k )) of this sequence with λ 1 (m k ) =0for each k, and 1 λ 1 (m k ) → 0 (as k →∞). Thus z 1 + λ 2 (m k ) λ...

Ngày tải lên: 04/07/2014, 10:20