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511-Final-2008F

# 511-Final-2008F - x there is an open segment of the line...

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Economics 511 Professor John H. Nachbar Fall 2008 Final You have two hours. Write your answers clearly, with good penmanship and good syntax. A “correct” but unintelligible answer is a wrong answer. 1. Prove the following. Theorem. Let ( X, d x ) be a metric space. If a set C X is complete and totally bounded then it is sequentially compact. 2. Prove the following. Theorem. Let X and Y be vector spaces. If L : X Y is linear then dim( K ( L )) + dim( L ( X )) = dim( X ) . 3. Recall that, given metric spaces ( X, d X ) and ( Y, d Y ) and a function f : X Y , define f to be continuous at x * X iff for any ε > 0, setting y * = f ( x * ), N δ ( x * ) f - 1 ( N ε ( y * )). Prove the following. Theorem. Let ( X, d X ) and ( Y, d Y ) be metric spaces. f : X Y is continuous at x * X iff, setting y * = f ( x * ) , lim x x * f ( x ) = y * . 4. Prove the following. Theorem. For any A [0 , 1] , if λ * ( A ) = 0 then A is measurable. 5. Given a vector space X (possibly infinite dimensional) and a set A X , a point x * X is internal to A iff for every x X , x 6 = x * , there is a real number ε > 0, which can depend on x , such that for any real number θ N ε (0), x * + θx A . In words, a point is internal iff, for any line through

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Unformatted text preview: x * , there is an open segment of the line that (a) contains x * and (b) is a subset of A . The length of the segment can depend on the line. The property of being internal is related to, but distinct from, the property of being interior . Recall that if ( X,d ) is a metric space and A ⊆ X , then x * is interior to A iﬀ there is an ε > 0 such that N ε ( x ) ⊆ A . (a) Provide an example in R 2 of a set A and a point x * such that x * is internal to A but not interior to A . A picture with a clear accompanying explanation is ideal. (b) Prove that if x * is interior to A ⊆ R N then it is internal to A . (c) Consider ( R ∞ ,d p ), where d p is the pointwise convergence metric. Is the point (1 , 1 ,... ) internal to R ∞ + ? If yes, provide a proof; if no, provide a concrete, clear counterexample....
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511-Final-2008F - x there is an open segment of the line...

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