UW-Madison MATH 521 - Assignment08 (2 pages)

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Assignment08



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Assignment08

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School:
University of Wisconsin, Madison
Course:
Math 521 - Analysis I

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MATH 521 Spring 2014 Assignment 8 Due date Monday May 5 Submit only the starred questions Honors students submit the Honors Question as well 1 Suppose f a b 7 R is bounded on a b Prove that if f is Riemann integrable on a b then the function Z x f t dt F x a is uniformly continuous on a b R2 2 Suppose f 0 2 7 R is continuous on 0 2 and that 0 f t dt 2 Show that there is R xa c 0 2 such that f c c Hint Consider the function F x 0 f t t dt and apply the FTC 3 Use the definition of Riemann integrability or an equivalent formulation to prove that Z b 1 2 x dx b a2 2 a 4 Suppose that f g a b 7 R are Riemann integrable on a b Define the following s Z b kf k2 f x 2 dx a Z hf gi b f x g x dx a There are the basis for the L2 metric space on functions Note that the objects in this space are functions rather than vectors as they were in Rn Nevertheless we can see immediate parallels with the Euclidean norm k k2 and dot product which is often written in inner product notation h i a Prove directly that if f x is Riemann integrable on a b then f x 2 is Riemann integrable on a b This guarantees that we may evaluate kf k2 b Compute kfn k2 n N for the family of functions for 1 n x 0 1 nx 1 nx for 0 x 1 n fn x 0 otherwise on the interval 1 1 What happens as n Does this make sense in the context of the individual functions fn x Are there are peculiarities Note The norm is a measure of distance of a function from the trivial zero function f x 0 c Prove that hf gi kf k2 kgk2 Hint It will help to look back at the proof of the Cauchy Schwarz Inequality contained in the Week 4 notes Consider starting with the identity Z bZ a b f x g y f y g x 2 dy dx 0 a d Prove that kf gk2 kf k2 kgk2 Hint use c Honors Define the set L2 a b to be the set of all functions which are Riemann integrable on a b Define d L2 a b L2 a b 7 R according to d f g kf gk2 Show that d f g satisfies all the metric space condition except d f g 0 implies f x g x There are subtleties with this final metric space condition



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