The Metric Spaces[Examples and Counter Examples]

Metric spaces

Let X be a non-empty set,a metric $d$ on X is a function
$d:X\times{X}\rightarrow\mathbb{R}$ which satisfy the following distance properties:
for all $x,y,z\epsilon{X}$
(i)$d(x,y)\geq0$
(ii) $d(x,y)=0$
(iii) $d(x,y)=d(y,x)$
(iv) $d(x,z)=d(x,y)+d(y,z)$

Examples of Metric Spaces

(1)The Euclidean space: this is a metric space defined on the set of n-tuples of
real numbers $\mathbb{R}^{n}$ in this space we use
$d(x,y)=(\sum^n_{i=1}|x_i-y_i|)^\frac{1}{2}$ for all
$x_i,y_i\in{X}$ using the axioms of metric spaces,the distance function defined on
the euclidean space will satisfy the metric properties, but remember we might have
to use some inequalities to establish our proof such as the cauchy schwarz inequality.
(2)The Minkowsky Metrics:the minkowsky metric defined on the set of n-tuples
$\mathbb{R}^{n}$. for a non-empty set X and $x,y\in\mathbb{R}^{n}$ then
$x=(x_1,x_2,x_3,...)$ and $y=(y_1,y_2,y_3,..)$ for any $1\leq{p}<\infty$ we define a
distance $d(x,y)=(\sum^n_{i=1}|x_i-y_i|^p)^\frac{1}{p}$ which satisfy the metric property.
(3) the set of all sequence spaces e.g the $L_\infty$ spaces and the $L_p$ spaces.
(4) the function spaces

Counter example

for all $x,y\in{X}$ and $1\in\mathbb{R}$
(1)the metric $d(x,y)=\frac{1}{|x+y|}$ is not a metric as it does not satisfies all
properties of metric spaces.
(2)the metric $d(x,y)=\frac{d(x,y)}{1+d(x,y)}$ is not a metric space as it does not
satisfy all properties of metric spaces.