MATH 421/510
HW 1
Due January 28, 2016 (in class)
1. Chapter 5, Exercise 6
2. Chapter 5, Exercise 7
3. Chapter 5, Exercise 12
4. We saw in Chapter 5, Exercise 6(c) that the closed unit ball is compact
for a finite dimensional normed vector space. In this question, we
investigate the question in infinite dimensional setting. Let X is be an
infinite dimensional normed vector space. Show that the closed unit
ball centered at 0 is not compact. (Hint: See Chapter 5, Exercise 19(a).
You may use one of the three equivalent defintions of compactness given
in Theorem 0.25.)
5. Chapter 6, Exercise 5