Macroeconomics Exam Review 163

Macroeconomics Exam Review 163 - 2 Let and be nonempty...

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Unformatted text preview: 2. Let and be nonempty, disjoint, convex subsets in a normed linear space with compact and closed. By Exercise 1.208, there exists a convex neigh- borhood ∋ such that ( + ) ∩ = ∅ . By the previous part, and can be strongly separated. 3.197 Assume ( , ) = inf {∥ x − y ∥ : x ∈ , y ∈ } = 2 > 0. Let = ( ) be the open ball around of radius . For every x ∈ , u ∈ , y ∈ ∥ x + ( − u ) − y ∥ = ∥ x − y − u ∥ ≥ ∥ x − y ∥ − ∥ u ∥ so that ( + , ) = inf x , u , y ∥ x + ( − u ) − y ∥ ≥ inf x , u , y ( ∥ x − y ∥ − ∥ u ∥ ) ≥ inf x , y ∥ x − y ∥ − sup u ∥ u ∥ ) = 2 − = > Therefore ( + ) ∩ = ∅ and so and can be strongly separated. Conversely, assume that and can be strongly separated, so that there exists a convex neighborhood of...
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