Homework 13: Metric Spaces

Instructions. Assignments should be stapled and written clearly and legibly. Problem 5 is optional.

1. Section 5.1, #5.6, 5.8.
2. Section 5.3, #5.25.
3. Section 6.1, #6.2.
4. (Linear Algebra) In this problem you will prove that the Euclidean metric \(d\) on \(\bR^n\) is a metric. For \(\bx,\by\in \bR^n\) and \(c\in \bR\), define \(\bx+\by\), \(c\bx\), and \(\bx\cdot \by\) to be the standard vector addition, scalar multiplication, and dot product respectively.
   1. Prove that \(\bx\cdot (\by+\bz) = (\bx\cdot \by)+(\bx\cdot \bz)\).
   2. Prove the Cauchy-Schwartz inequality: \(|\bx\cdot \by|\leq \|\bx\|\|\by\|\). (Hint: If \(\bx,\by\neq\mathbf{0}\), let \(a=1/\|\bx\|\) and \(b=1/\|\by\|\), and use the fact that \(\|a\bx\pm b\by\|^2\geq 0\).)
   3. Prove that \(\|\bx+\by\| \leq \|\bx\| + \|\by\|\). (Hint: Compute \(\|\bx+\by\|^2\) and apply (b).)
   4. Verify that \(d\) is a metric.
5. Section 5.3, #5.23.