Aktosun, Fall 2023, 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
    Last modified: April 2, 2023