Midterm_Review - Midterm Review Math 321 Spring 2015 1 Give...

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Midterm Review - Math 321, Spring 2015 1. Give an example of an equicontinuous family of non-constant functions that is not totally bounded. Sketch of solution. The function class Lip K [0 , 1] \ { constant functions } for any fixed K pro- vides an example. This family is equicontinuous because the continuity parameter δ can be chosen to be /K independent of the functions in this class. On the other hand, any totally bounded set must be bounded, whereas Lip K ([0 , 1]) contains the unbounded collection of all functions of the form x + C , with C an arbitrary real number. 2. Find a uniformly convergent sequence of polynomials whose derivatives are not uniformly convergent. Sketch of solution. The sequence p k ( x ) = x k +1 / ( k + 1) is uniformly convergent because || p K || 1 k +1 0. On the other hand, p 0 k ( x ) = x k , which converges uniformly to the function q which takes the value 1 at x = 1 and is zero elsewhere. Since each p 0 k is continu- ous and q is not, the sequence { p 0 k } cannot be uniformly convergent.
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