《实分析博士资格考试题real analysis》.pdfVIP

Ph.D. Qualifying Examination Sem 2, 2002/2003 Analysis 1 [20 marks] a Let f : R → R be a di?erentiable function. If f ?1 2 and f 1 2, show that there exists x ∈ ?1, 1 such that f x 2. Hint: consider the function 0 0 f x ? 2x and recall the proof of Rolle’s theorem b Let f : ?1, 1 → R be a di?erentiable function on ?1, 0 ∪ 0, 1 such that limx→0 f x l. If f is continuous on ?1, 1 , show that f is indeed di?erentiable at 0 and f 0 l. 2 Let Pn be the space of polynomials of degree ≤ n on R for each n ∈ N. If p x a + a x + a x2 + · · · + a xn , de?ne [20 marks] 0 1 2 n ||p || max |a |, |a |, · · · , |a | M 0 1 n ||p ||∞ max |p x | : x ∈ [0, 1] , and ||p ||1 01 |p x |dx. i Show that || · ||1 is a norm of the space Pn . ii Use the fact that || · ||M and || · ||∞ are also norms of Pn , or otherwise, to show that there exists a positive constant cn such that c ||p || ≤ ||p || ≤ 1/c ||p || n ∞ 1 n M for all p ∈ Pn . Hint: note that Pn ≡ Rn+1 iii With the help of the Weiestrass approximation theorem, show that there is no positive constant c such that cn c for all n. Note that for each ε 0, there is a nonnegative continuous function f on [0, 1] such that f 0 1 and ||f || ε. ε ε ε 1 2 3 Prove or disprove each of the following statements. [40 marks] a If f : [1, 5] → [1, 5] is a continuous function, then there exists x0 ∈ [1, 5] such that f x x . 0 0 b Let f be a sequence of uniformly continuous functions on an interval I . If f n n converges uniformly to a function f on I , then f is also uniformly continuous on I . c Let f be a sequence of functions that converges uniformly to a function f on n 0,2 . If each of the fn is di?erentiable on 0,2 , then f is also di?erentiable on 0,2 . d If f is a continuous function on [-1,1], then there exists a constant M 0 such that |f x ? f