# Aktosun, Fall 2022, Math 5327, Sample Problems 2

1. Determine if the following are metrics on the set of real numbers:
1.     d(x,y)=|x-y|
2.     d(x,y)=|x-y|2
3.     d(x,y)=|x-y|1/2
4.     d(x,y)=|sin(x)-sin(y)|
5.     d(x,y)=|ex-ey|
6.     d(x,y)=|x-y|/(|x+y|)
7.     d(x,y)=|x-y|/(1+|x+y|)
8.     d(x,y)=√2 |x-y|
9.     d(x,y)=[|x-y|/(|x+y|)]1/2
10.     d(x,y)=|sin(x)-sin(y)|
11.     d(x,y)=|√|x|-√|y||
12.     d(x,y)=√|x|-|y|

2. Determine if the following are norms on the set of real numbers:
1.     |x|
2.     |x|2
3.     |x|1/2
4.     |x|3/2
5.     |sin(x)|
6.     |ex|
7.     |x|(1+|x|)
8.     |x|/(1+|x|)
9.     √2 |x|

3. Determine if the following are inner products on the set of real numbers:
1.     ⟨x,y⟩=sin(x)-sin(y)
2.     ⟨x,y⟩=xy
3.     ⟨x,y⟩=cos|x-y|
4.     ⟨x,y⟩=|xy|
5.     ⟨x,y⟩=|x| y

4. Describe the following with appropriate notation:
1.     The metric space of real-valued continuous functions on the interval [a,b] with a metric related to the Lp norm
2.     The metric space of real-valued continuous functions on the interval [a,b] with a metric related to the sup norm

Tuncay Aktosun aktosun@uta.edu