Mathematics – selected topics
Set no 4
Problem 1
Let (x1,y1) and (x2,y2) denote two arbitrarily selected elements of the space R2 , such that
points (x1,y1), (x2,y2) and (0,0) are not co-linear. Prove that elements (x1,y1) i (x2,y2)
constitute a basis in R2.
Problem 2
Let a, b  R be a closed interval. Consider a set of functions
b


2
L a, b :   f :a, b  R :  f  x  dx   .
a


(square-integrable functions).
2
a) Assume that f , g  L2 a, b . Prove the following inequality (the Schwarz
inequality):
2
b
 b 2
 b 2


  f t g t dt     f t dt     g t dt  .
a
 a
 a

b) By means of the Schwarz inequality prove that L2 a, b is a linear space.
Problem 3
Let (X, d) be a metric space.
 An open ball with a center x0 and a radius r > 0 is the following set:
Bx0 , r  :  x  X : d ( x, x0 )  r
 A closed ball with a center x0 and a radius r > 0 is the following set:
BC x0 , r  :  x  X : d ( x, x0 )  r
 A ring with a center x0 and a radii r2 > r1 > 0 is the following set:
Rx0 , r1 , r2  :  x  X : r1  d ( x, x0 )  r2 
Prove that the following functions are metrics in X. Find Bx0 , r  , BC x0 , r  , Rx0 , r1 , r2 
in each case.
a) X is a given non-empty set.
 : X  X  0,,
x y
x y
0 , if
1 , if
  x, y   
(so-called discrete metric).
b) X = R2
 : R2  R2  0,,
 x1, y1 , x2 , y2   x2  x1  y2  y1
(so-called taxi driver metric)
c) C (complex numer set)
 : C  C  0,,
 z1, z2   z1  z2 , where z denote modulus of a complex
number z.
Problem 4
a) Let x1 ,..., xn  and  y1 ,..., yn  be two elements of the space Rn.
Prove the following inequality, which is called the Cauchy inequality:
2
 n
  n   n

  xk yk     xk2     yk2  .
 k 1
  k 1   k 1 
b) Using the above inequality prove that
 x1 ,..., xn ,  y1 ,..., yn  
is a metric in
Problem 5
Let X , 
n
y
k 1
k
 xk 
2
Rn
 be a normed space. Prove that function
 x, y   x  y
is a metric in X.
Problem 6
Let x1 ,..., xn  be an element of Rn space. Prove that the function
x1,..., xn 
 x12  x22  ...  xn2
is a norm in Rn .
Problem 7
Let f  L2 a, b . Prove that the function
b
f 
 f t dt
2
a
is a norm in L2 a, b.
Problem 8
Let X , 
 be a normed space. Assume that lim x
lim xn  x0 .
n 
n
n
 x0 . Prove that