Real Analysis with Economic Applications - Chapter C pps

... finite? Exercise 28. H (a) Show that a separable metric space need not be compact, but a compact metric space X is separable. (b) Sho w that a connected metric space need not be compact, and a compact ... a metric space as compact. This is mostly due to the fact that some textbooks which focus exclu- sively on Euclidean spa c es define compactness as the combination o f the properties of clos...

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... yourself in the case where X is a compact interval. 176 3.2 The Local-to-Global Method Since any compact set is bounded (Proposition C. 5), Corollary 2 is really an immedi- ate c on sequence of Proposition ... ?) Conclusion: X is not connected.  In the previous chapter w e were una ble to provide many examples of c onnected sets. We c haracterized the c lass of all connected sets...

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... our intuitive discussion above that forces us to conclude that S is not a set; it is a collect ion of objects (sets in this case) which is considered asasingleentity. ButwecannotacceptS asa“set,”forifwedo,wehavetobeable to ... R T is decreasing or strictly decreasing, if −f is increasing or strictly increasing, respectively. By a m onotonic function in R T , w e understand a map in R T which...

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... indifference and indecisiveness in general.) A t the v ery least, there are in- stances in which a decision maker is actually composed of sev eral agents eac h w ith a possibly distinct objective function. ... 1]. 80 Chapter B Countabilit y This chapter is about Cantor’s coun tability theory whic h i s a standard prerequisite for eleme ntary real analysis. Our tre atment is incom p lete...

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... is con vex, the n ϕ ∗ is a concave and c ontinuous function. Exercise 40. (Robinson-Day) Let Θ be any metric space, Γ : Θ ⇒ R be a continuous, compact- and convex-valued correspondence, φ ∈ C( R ... metric on the class of all nonempty compact subsets of X. Nadler’s Contractiv e Correspondence T heorem. Let X be a complete metric space, and Γ a compact-valued self-correspondence on X.IfΓ is...

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... joint cost of a comm on facility. (For concreten ess, y ou may view each of these agents as coun tries / cities and the common facilit y as a joint irrigation/h ydroelectric power system. Such scenarios ... a reasonab le cost shari ng rule φ that maps a cost allocation problem c to a vector (φ 1 (c) , , φ |N| (c) ) ∈ R |N| + such that S i∈N φ i (c) =c( N) with φ i (c) being the cost a...

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... (Example E.2) is a convex-valued corre- spondence which is not convex. (d) Ev ery linear correspondence Γ : X ⇒ Y is convex (Exercise F.31). Is the con- verse true? (e) (Deutsch-Singer) Prove: If Γ ... statemen t “closed halfspaces” with “open half- spaces” in Proposition 2? (b) Can one omit the condition al-in t X (S) = ∅ in Proposition 2? Exercise 50. Let C be an algebraically closed c...

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... pessimistic person would behave. (It should be noted that the c hoice of a particular set of beliefs in an application is a complicated matter which is best d iscussed within the specifics of that ... that will obtain in the future. In financial economics, such an act is th us called a state-contingent claim,andwe shall a dh e re to this terminology in this section as we ll. As is large ly st...

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... the closure of which is compact in X. O bv iously, every compact met- ric space is locally compact, but not con ve rsely. (For instance, R is locally comp act, while, of cou rs e, it is not compa ... metric d with the group st ructu re of the lin ear 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 dist...

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... ∈ C( T ) when T is compact), what we ha ve in mind is the sup-norm of this function , that is, f ∞ . Similarly, we norm c 0 , c 0 and c by the sup-norm, and hence c 0 is subspace of c 0 , c 0 is ... (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 wi...

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