2.2 KiB
Metric spaces
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 of metric spaces
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_{j=1}^n (x(j) - y(j))^2},
for all x,y \in \mathbb{R}^n
with x = (x(j))
and y = (y(j))
. 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)
.
??? note "Proof:"
Will be added later.
Definition 2: let
l^p
withp \geq 1
be the set of sequencesx \in l^p
of complex numbers with the property that
\sum_{j \in \mathbb{N}} | x(j) |^p \text{ is convergent},
for all
x \in l^p
.
We have that a metric d
for l^p
may be defined by
d(x,y) = (\sum_{j \in \mathbb{N}} | x(j) - y(j) |^p)^\frac{1}{p},
for all x,y \in l^p
.
??? note "Proof:"
Will be added later.
From definition 2 the sequence space l^\infty
follows, which is defined as the set of all bounded sequences x \in l^\infty
of complex numbers. A metric d
of l^\infty
may be defined by
d(x,y) = \sup_{j \in \mathbb{N}} | x(j) - y(j) |,
for all x, y \in l^\infty
.
??? note "Proof:"
Will be added later.