homework_2aN - Homework 2 Math 104 A 1 Instructor Prof Hector D Ceniceros General Instructions Please write your homework papers neatly You need to turn

homework_2aN - Homework 2 Math 104 A 1 Instructor Prof...

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Homework # 2 , Math 104 A 1 Instructor: Prof. Hector D. Ceniceros General Instructions : Please write your homework papers neatly. You need to turn in both full printouts of your codes and the appropriate runs you made. Write your own code, individually. Do not copy codes! 1. Let V = R 2 . Sketch the unit ball for the norms k · k 1 , k · k 2 , and k · k . 2. We say that a sequences of functions { f n } defined on [ a, b ] converges uniformly to a function f if for each > 0, there is N , which depends only on and [ a, b ] but is independent of x , such that | f n ( x ) - f ( x ) | < , if n > N , for all x [ a, b ] . (1) Define the sequence of numbers M n = k f n - f k . Prove that { f n } converges uniformly to f in [ a, b ] if and only if M n converges to zero as n → ∞ . 3. (a) Prove that the sequence of functions given by f n ( x ) = n - 1 n x 2 + 1 n x, 0 x 1 (2) converges uniformly to f ( x ) = x 2 in [0 , 1]. (b) Does the sequence f n ( x ) = x n defined in [0 , 1] converge uniformly? 4. Let f ( x ) = ( x for 0 x 1 / 2 , 1 - x for 1
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