This document defines and provides examples of metric spaces. It begins by introducing metrics as distance functions that satisfy certain properties like non-negativity and the triangle inequality. Examples of metric spaces given include the real numbers under the usual distance, the complex numbers, and the plane under various distance metrics like the Euclidean, taxi cab, and maximum metrics. It is noted that some functions like the minimum function are not valid metrics as they fail to satisfy all the required properties.

1. Pratap College Amalner
T. Y. B. Sc.
Subject :- Mathematics
Metric Space
Prof. Nalini S. Patil
(HOD)
Mob. 9420941034, 9075881034

2. Metric Space

3. Metric Space
• Introduction
• Definition
• Example

4. Introduction
• Definition Absolute-value function on R
Let f : R R be a function defined by
f( x) = |x|= x if x > 0
= -x if x < 0
= 0 if x = 0
• Properties of absolute value function
i) |0| = 0
ii) |a| > 0 if a ≠ 0
iii) |a| = |-a|
iv) |a+b|≤ |a|+|b|

5. Introduction
• Now, for x, y real numbers, the geometric
interpretation of |x-y| is the distance from x
to y. If we define the “distance function” ρ by
ρ(x, y) = |x-y|
then the properties i) – iv) have the following
consequences for any points x, y, z real
numbers

6. Introduction
v) ρ(x, x) = |x-x| = 0
(i.e, the distance from a point to itself is 0)
vi) ρ(x, y) > 0 (x ≠ y)
(The distance between two distinct points is
strictly positive)
vii) ρ(x, y) = ρ(y, x)
(The distance from x to y is equal to the
distance from y to x.)

7. Introduction
viii) ρ(x, y) ≤ ρ(x, z) + ρ(z, y)
( triangle in equality)
Distance function satisfies v) – viii) properties
A Distance function is usually called a metric

8. Definition
• Definition : Let M be any set. A metric for M is
a function ρ with domain M M and range
is contained in [0, ) such that for any
element x, y, z of set M
ρ(x, x) = 0
ρ(x, y) > 0 (x ≠ y)
ρ(x, y) = ρ(y, x)
ρ(x, y) ≤ ρ(x, z) + ρ(z, y)

9. Definition
• If ρ is metric for M, then the order pair (M, ρ)
is called Metric Space.
• A metric for M thus has all properties v)-viii) of
the distance function |x - y| for R
• Example : f : R2 R be a function defined by
f(x, y) = |x - y|.
Then f is metric for R, this metric is known as
absolute vale metric

10. Example
 The discrete metric is defined by the formula
d(x, y) = 1 if x ≠ y
= 0 if x = y
• By definition of d, d(x, x) = 0
• If x ≠ y then d(x, y) = 1 > 0
• Clearly d(x, y) = d(y, x)
• triangle inequality d(x, y) ≤ d(x, z) + d(z, y).
If x = y, then the left hand side is zero and the
inequality certainly holds. If x ≠ y , then the left hand
side is equal to 1. Since x ≠ y , we must have either
x ≠ y or else x ≠ y . Thus, the right hand side is at least 1
and the triangle inequality holds in any case.

11. Examples
The plane R2 with the "usual distance"
(measured using Pythagoras's theorem):
d((x1 , y1), (x2 , y2)) = squ[(x1 - x2)2 + (y1 - y2)2].
This is sometimes called the 2-metric d2 .
• This is also metric
space for R2
z= (x1 , y1) w = (x2 , y2)

12. Example
The same picture will give metric on the
complex numbers C interpreted as the Argand
diagram. In this case the formula for the
metric is now:
d(z, w) = |z - w|
where the | | in the formula represent the
modulus of the complex number rather than
the absolute value of a real number.

13. Example
The plane with the taxi cab metric
d((x1 , y1), (x2 , y2)) = |x1 - x2| + |y1 - y2|.
This is often called the 1-metric d1 .

14. Example
The plane with
the supremum or maximum metric
d((x1 , y1), (x2 , y2)) = max(|x1 - x2|, |y1 - y2| ).
It is often called
the infinity metric d .

15. Example
• To understand them it helps to look at the unit
circles in each metric.
That is the sets { x belongs to R2 | d(0, x) = 1 }.
We get the
following picture:

16. Example
 If ρ is metric on M
then 7ρ is also metric on M
If ρ and f are metric on M then ρ+f is also
metric on M
 But -7ρ is not metric on M.
Since second property of non-negativity not
satisfied by ρ

17. Question
• Set of all metric on set M is Group?

18. Example of Not Metric
ρ : R2 R be a function defined by
ρ(x, y) = 0
then ρ is not metric
Since 1 ≠ 2 but ρ(1, 2) = 0 (by def. of ρ)
other properties are satisfied

19. Example of Not Metric
The plane with the
minimum function
d((x1 , y1), (x2 , y2)) = min(|x1 - x2|, |y1 - y2| ).
• Then d is not metric
Since d((1, 2),(1,5)) = min(|1-1|,|2-5|)
= min (0, 3) = 0
Second property not Satisfied.