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Math 131B midterms

# Math 131B midterms - Math 131b-Midterm Name 1(25 points...

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Math 131b-Midterm Name: 1. (25 points) Indicate whether each of the following statements is true or false . A correct response will receive full credit. An incorrect response may receive partial credit if accompanied by a reasonable explanation. Read each question carefully. (a) The space C [0 , 1] of continuous functions on [0 , 1] with norm given by f = sup {| f ( x ) | , x [0 , 1] } is a complete normed linear space. Solution. True. (b) If { f n } is a sequence of functions converging uniformly to a function f on a closed interval [ a, b ] , then f n f pointwise. Solution. True. (c) If { f n } is a sequence of continuous functions converging pointwise to a function f on a closed interval [ a, b ] , then f n f uniformly. Solution. False. (d) The sequence of functions f n ( x ) = x n on the open interval (0 , 1) converges uniformly to 0 . Solution. False. We know that on [0 , 1] , f n ( x ) f ( x ) with f ( x ) = 0 for x = 1 and f (1) = 1 . Since the f n ( x ) are continuous but f isn’t, it follows that { f n } does not converge uniformly on [0 , 1] . Now suppose f n f uniformly on (0 , 1) . Then for every there is a N such that n > N, x (0 , 1) ⇒ | f n ( x ) - f ( x ) | ≤ . On the other hand, since f n (1) = 1 = f (1) , f n (0) = 0 = f (0) for all n , it would follow that actually n > N, x

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