508F06Ex1solns

508F06Ex1solns - Math 508 Exam 1 Jerry L. Kazdan October...

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Unformatted text preview: Math 508 Exam 1 Jerry L. Kazdan October 12, 2006 12:00 1:20 Directions This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5 points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems (60 points, 15 points each). Closed book, no calculators but you may use one 3 00 5 00 card with notes. Part A: Examples (4 problems, 5 points each). Give an example of an infinite set in a metric space (perhaps R ) with the specified property. A1. Bounded with exactly two limit points. Solution: The set { (- 1) n (1 + 1 n ) , n = 1 , 2 , 3 , . . . } in R . A2. Containing all of its limit points. Solution: Lots of exmples: 1). The empty set. 2). All of R . 3). The point { } R . 4). The closed interval { x 1 in R } . A3. Distinct points { x j , j = 1 , 2 , 3 , . . . } with x i 6 = x j for i 6 = j that is compact. Solution: The following subset of the real numbers: { } { 1 n , n = 1 , 2 , 3 , . . . } . A4. Closed and bounded but not compact. Solution: The closed unit ball k x k 1 in 2 . The standard basis vectors e 1 = (1 , , , . . . ), e 2 = (0 , 1 , , , . . . ), etc have no convergent subsequence....
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This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.

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508F06Ex1solns - Math 508 Exam 1 Jerry L. Kazdan October...

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