solutionshw6-201141

solutionshw6-201141 - Solutions for HW 5 Problem 107: 2....

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Solutions for HW 5 Problem 107: 2. Let M = | | + | | 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 | f n ( x )+ g n ( x )- f ( x )- g ( x ) | | | sup x | f n ( x )- f ( x ) | + | | 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 ....
View Full Document

Page1 / 2

solutionshw6-201141 - Solutions for HW 5 Problem 107: 2....

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online