notes/docs/mathematics/functional-analysis/metric-spaces/metric-spaces.md

Metric spaces

Definition 1: a metric space is a pair (X,d), where X is a set and d is a metric on X, which is a function on X \times X such that

1. d is real, finite and nonnegative,
2. \forall x,y \in X: \quad d(x,y) = 0 \iff x = y,
3. \forall x,y \in X: \quad d(x,y) = d(y,x),
4. \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 with p \geq 1 be the set of sequences x \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.