solutionshw6-201141

# solutionshw6-201141 - Solutions for HW 5 Problem 107 2 Let...

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Unformatted text preview: Solutions for HW 5 Problem 107: 2. Let M = | α | + | β | and let > 0. Then there exists N such that sup x | f n ( x )- f ( x ) | < 2 M for all n ≥ N and sup x | g n ( x )- g ( x ) | < 2 M for all n ≥ N . Now sup x | αf n ( x )+ βg n ( x )- αf ( x )- βg ( x ) | ≤ | α | sup x | f n ( x )- f ( x ) | + | β | sup x | g n ( x )- g ( x ) | < 2 + 2 = for all n ≥ N . The product f n g n does not have to converge uniformly to fg . Example: Let X = R and f n ( x ) = g n ( x ) = x + 1 n . Then f n converges uniformly to f , where f ( x ) = x . On the other hand sup x | f n ( x ) 2- x 2 | = sup x | 2 x n + 1 n 2 | ≥ 2 (put x = n ) for all n . Remark: If both f and g are bounded on X , then also f n and g n will be uniformly bounded from some n on, when they converge uniformly to f , respectively g . Then one can show, by adding and subtracting the term f n g , that also the product f n g n converges uniformly to fg ....
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solutionshw6-201141 - Solutions for HW 5 Problem 107 2 Let...

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