MATH 450 Final: MATH 450 Amherst S10M42Final

• School: Amherst College
• Department: Mathematics and Statistics
• Course: MATH 450
• Professor: All

Document Summary

Please answer each problem as clearly and completely as you can. You may use royden and your notes and problem sets, but no other books or sources, and you should not discuss the problems with anyone except me. Problem 1 suppose 0 < 1. A function f : [0, 1] r is said to be h older continuous of exponent if there exists a constant m > 0 such that |f (x) f (y)| |x y| m for all x, y [0, 1] with x 6= y. Let be the set of all h older continuous functions on [0,1] of exponent , and de ne kf k to be kf k = |f (0)| + sup x6=y. Prove that kf k is a norm which makes into a banach space. (remark: = 1 yields. Problem 2 use a convergence theorem to directly prove that n z lim.