Real Analysis with Economic Applications - Chapter A pot

... pretty ric h algebraic structure. In particular, it is a partially ordered linear space (see Chapter F). Wh en the dom ain of a real function is a poset, we can talk about how this map a ects the ... exists a function f : A → V A such that f (A) ∈ A for each A ∈ A. One can reword this in a few other wa y s. 12 But, how about the following algorithm? Start with A 1...

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... an arbitrary function, i =1, , m. This sort of a game i s called an aggregative game. 28 As we shall see below, man y in ter- esting economic games are aggregative. (See Corchó n (1996) for a ... ,m } in game theory; we also adopt this no- tation below.) We say that a strategic game G is finite i f e ach action space X i is finite. By far t he most f amous example o f a s trategic gam...

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... A real- valued linear operator is r eferred to as a linear fun ct ion a l on X. 17 Notation. T h e s et of all linea r o perators from a linear sp ace X in to a line ar space Y is denoted as ... ∈ N and fixsomeA ∈ 2 N \{∅}. The capacity u A ∈ V N defined as u A (B):=  1 ,A B 0 ,A- B (15) is called a unan im ity gam e . We can again think of this game as corresponding to a vo...

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... arbitrage a ssociated with a p rice functional π is a state- contingent claim a ∈ R Ω such tha t a | 0 and π (a) ≤ 0, or a  0 and π (a) < 0. Thus, w e say that the price functional π is arbitrage-free ... contains a d ollars and a state-contingent claim a. By a standard agen t f or Ω in t he presen t s etting we me an a concav e function u : Z Ω → R such that b> ;...

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... e ntable in view of Ex ercise 24. Tak e any real a, an d p ick any z a ∈ Y such that (a, 1)  lex z a and z a  lex (a, 0). Clearly, we mu st have z a 1 = a and z a 2 ∈ (0, 1). But then z a = ... Then, for any a ∈ R,wehave u (a, a +1)>u (a, a) so that I (a) :=(u (a, a) ,u (a, a +1))is a nondegenerate interval in R.Moreover,I (a) ∩ I(b)=∅ for any a = b, for w...

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... speak of  p as a metric space, we always have this metric in m in d !) O f course,wehavetocheckthatd p is well-defined as a real- valued function, and that it satisfies the triang le inequ ality. ... multidimensional analogue of the bisection argum ent given above. In Section 4 w e will give an alter- native proo f. Exercise 26. Let n ∈ N, and take any real numbers a and b with b>...

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... For any real numbers a and b with a& lt;b,show that C[0, 1] and C [a, b] are isometric. Exercise 18. Let d and D betwometricsonanonemptysetX. (a) Show that d and D are equivalent iff id X is an homeomorphism. (b)(Carothers) (X, ... viewed as a functional equation in w hich the unkno w n variable is a function — it is often ca lled Cauchy’s (first) fun c tio n al equat io n. 25 In ge...

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... i functionals? T he main reason is that these functionals g iv e us a n analytic charac- terization of an arbitrary algebraically closed convex set with nonempt y a lgebraic 346 [4] The linear operator D : ... id es a novel application of Proposition 4 to linear algebra. E{dpsoh 13. (Farkas’ L emma)Takeanym ∈ N, and any u 1 , , u m ,v ∈ R n . A famou s result of linear algebra , Far...

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... space X. Translation in variance is thus a concept whic h is partly met- ric a nd partly algebraic — it connects in a particular w ay the distance function o n a linear space with the operation of ... functional analysis and operator theory in a substantial way — his contributions have accen tuated the development of quantum mechanics early in the 20th century. There are at leas...

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... not have a Schauder basis; (e)If X has a countable Schauder basis, then it i s separable. Remark 1. One major reason wh y an n-dim e ns ional linea r space is a well-behaved entit y is th at there ... think of an infinite dimensional Banach space as a “large” n orm ed linear space. In particular, an y  p is, intuitively, “much larger” than any of its subspaces that are spanned by coun...

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