Real Analysis with Economic Applications - Chapter D ppsx

... homeomorphism. (b)(Carothers) (X, d) and (X, D) may be homeomorphic, even if d and D are not equivalent. For instance, let X := {0, 1, 1 2 , 1 3 , } ,d: = d 1 , and de fine D on X 2 as follows: D( x, 1) := x and D( x, 0) := ... functions definedonanarbitrarymetric space. Let T be an y metric space, and recall that B(T ) stands for the set of all bounded functions defined on T, whichi...

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... Q f,x (x+) is called the right-derivative of f at x, and is denoted by f  + (x). Similarly, if Q f,x (x−) ∈ R, then f is said to be left -d ifferentiable at x, and the left-derivative of f at ... called the least upper bound of S.) The lo wer bounds of S and inf S are defined dually. (The n umber inf S is called the greatest low er bound of S.) The main difference between Q and R is captured ....

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... said 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 ... countable and order-den s e loset with neith e r maximum nor mini- mumelementsisorder-isomorphictoQ (and, therefore, to Q ∩ (0, 1)). 3 SomeMoreAdvancedSetTheory 3.1 The Cardinality Ordering We...

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... X, and denote the classes of all open subsets of X with respect to d and D as O (d) and O (D) , respectively. We say tha t d and D (and/o r (X, d) and (X, D ))areequivalen t if O (d) =O (D) ,and that ... under e quivalence of metrics. Indeed, if (X, d) is an unbounded metric space, then (X, d 1 +d ) is a bounded metric space, whereas d and d 1 +d are equivalent. (Recall...

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... modeled by t he correspondence d : R n+1 ++ ⇒ R defined b y d( p, ι):=argmax{u(x):x ∈ B(p, ι)} . As you may reca ll, t he co rrespondence d is called t h e dem and correspondence of the individual. ... pro vided t hat A and B are bounded. (See Figure 3.) When d H is viewe d as a ma p on c(Y ) × c(Y ), it is called the Hau sdorff metric.(Note.Inthatcasesup can b e replaced with max in...

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... two operations defined on it. Two different t ypes of a dd ition and scalar mu ltiplication operations on a given set ma y well endow this set with different l inear structures, and henc e yield two v e r y differen ... r alization of the previous definitions. Indeed, by (6) and (5), for any x ∈ S, dim(S)=dim(aff (S)) = dim(aff (S) − x)=dim(span(S − x)). For instance, dim({x})=0and dim({λx +(1− λ)...

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... called the algebraic closure of S (in X), and is denoted by al-cl X (S). (Note. al-c l X (∅)=∅.)Ifal-cl X (S) ⊆ S, we say that S is algebr a ically closed in X. F in ally, the set al-cl X (S)\al-int X (S) ... S ⇒ R, defined by Γ(x):= {a ∈ R : a ≥ f(x)}, is a convex correspondence. (c) Show that the budget correspondence (Example E.2) is a convex-valued corre- spondence which is not convex. (d...

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... Axiom ∗ — called the C-Independence Axiom ∗ (“C” for “constan t”) — became the industry standard. The C-Independence Axiom ∗ . For any f, g ∈ H Ω,X ,p∈ L X , and any 0 < λ ≤ 1, f  g if and only if ... Gilboa-Schm eidler Multi-Prior Model In recen t years the theory of individual decision theory underwen t a considerable transformation, because the d escriptive po wer of its most foundat...

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... get d( w, Y )=0, which is impossible, given that w/∈ Y and Y is closed (Exercise D. 2). We conclude that (λ m ) is bounded. Now let λ := lim sup λ m . Then λ ∈ R (because (λ m ) is bounded) and there ... when endo wed with d ϕ , provided that ϕ −1 (0) = {0}. Th e following proposition collects some basic facts about met ric linear spaces that we will need later o n. It also provides a g...

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... tiable function s on [0, 1] with the sup-norm . Define D ∈ L(X,C[0, 1]) by D( f):=f  . Is D continuous? No. If i ∈ N, and f i ∈ R [0,1] is defined by f i (t):=t i , then f i  ∞ =1 and D( f i ) ∞ = i. (Why?) ... { D( f) ∞ : f ∈ B X } = ∞, so by Proposition 5, D is not bounded, and hence, not continuous. [4] The differen tia tion operator considered in [3] would be continuous if its...

Ngày tải lên: 04/07/2014, 10:20