1.4 KiB
Definition of a metric space
Definition 1: a metric space is a pair
(X,d)
, whereX
is a set andd
is a metric onX
, which is a function onX \times X
such that
d
is real, finite and nonnegative,\forall x,y \in X: \quad d(x,y) = 0 \iff x = y
,\forall x,y \in X: \quad d(x,y) = d(y,x)
,\forall x,y,z \in X: \quad d(x,y) \leq d(x,z) + d(y,z)
.
The metric d
is also referred to as a distance function. With x,y \in X: d(x,y)
the distance from x
to y
.
Examples
For the Real line \mathbb{R}
the usual metric is defined by
d(x,y) = |x - y|,
for all x,y \in \mathbb{R}
. Obtaining a metric space (\mathbb{R}, d)
.
??? note "Proof:"
Will be added later.
For the Euclidean space \mathbb{R}^n
with n \in \mathbb{N}
, the usual metric is defined by
d(x,y) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2},
for all x,y \in \mathbb{R}^n
with x = (x_i)
. Obtaining a metric space (\mathbb{R}^n, d)
.
??? note "Proof:"
Will be added later.
Similar examples exist for the complex plane \mathbb{C}
and the unitary space \mathbb{C}^n
.
For the space C([a,b])
of all real-valued continuous functions on a closed interval [a,b]
with a<b \in \mathbb{R}
the metric may be defined by
d(x,y) = \max_{t \in [a,b]} |x(t) - y(t)|,
for all x,y \in C([a,b])
. Obtaining a metric space (C([a,b]), d)
.