METRIC & METRIC SPACES : #metric #spaces #discrete #indiscrete #continuous #functions
https://youtu.be/lxZV_lBhtaw?si=PFQZ0TVgowSHvhTd
contents :
1. Metric Space.
2. Usual Metric
3. Discrete Metric
4. Indiscrete Metric
5. Metric on set of real valued continuous functions
6. Metric on Rn
Metric space:-
A metric on X is a real function d of ordered pairs (x, y) of elements of X which satisfies the following conditions.
(i) d(x, y)≥0 (non- negativity)
and d(x, y)=0 iff x=y.
(ii) d(x, y)=d(y, x) (symmetry)
(iii) d(x, y) ≤ d(x, z) +d(z, y) (transitivity)
Here the space X is called a metric space and it is denoted by (X, d) and the elements of X are called the points of X.
One should always keep in mind, however, that a metric space is not merely a non- empty set: it is a non-empty set together with a metric.
There are many different kinds of metric spaces, some of which play a very significant role in geometry and analysis. A few of them are
* Usual metric:-
A function d on the real line R is defined by
d(x, y)= |x-y| for any x, y Ꜫ R.
Is this d a metric?
Let us see
(i) d(x, y) = |x-y|
= distance between x and y
∴ d(x, y) ≥ 0
(ii) d(x, y) = |x-y|
= |-(y-x)|
= |-1||y-x|
= |y-x| ( |-1| = 1)
d(x, y)= d(y, x)
(iii) d(x,z) = |x-z|
= |x-y+y-z|
≤|x-y|+|y-z|
= d(x, y) + d(y,z)
∴ d(x,y) ≤ d(x,y) + d(y,z)
∴ The function 'd' on R by d(x,y)= |x-y |is a metric on R.
This metric d is called as Usual metric on R and hence The real line R is called the Usual metric space.
Similarly we can define a metric d on the complex plane C by for z1, z2 Ꜫ C,
d(z1, z2 )=| z1- z2 |
* Discrete Metric :
Let X be a non-empty set, and define d on X by d(x,y)= 0 for x=y and d(x,y) =1 for x ≠ y.
Clearly this function d is a metric on X and this metric is called a discrete metric on X and hence X is called as a discrete metric space.
* Indiscrete Metric :
A metric space X with the only open sets empty set and the entire space X itself is called an indiscrete metric space. An indiscrete metric space is also called as a trivial or co discrete metric space.
* Consider the nth dimensioal Euclidean space Rn ={( x1,x2,…,xn)/xi Ꜫ R}
This set is a metric space on which the metric d is defined as
for x = ( x1,x2,…,xn), y =( y1,y2,…,yn) Ꜫ Rn
d(x,y ) = sqrt( (x1-y1)2 +(x2-y2)2+…+(xn-yn)2 ).
* Let X be the set of all real valued continuous functions on [0,1]. This space X is also a metric space where the metric d on X is given by for any f,g Ꜫ X,
d(f,g) = Sup{|f(x)-g(x) |/x Ꜫ[0,1]}
People also ask :
1. What is the historical background of metric spaces?
#metric #spaces #discrete #indiscrete #continuous #functions
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