Definition:Distance/Points

Definition

Metric Space

Let $\struct {X, d}$ be a metric space.

The mapping $d: A \times A \to \R$ is referred to as a distance function on $A$ or simply distance.

Normed Vector Space

Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space.

Let $x, y \in X$ .

Then the function $\norm {\, \cdot \,} : X \times X \to \R$:

$\map d {x, y} = \norm {x - y}$

is called the distance between $x$ and $y$.

Real Numbers

Let $x, y \in \R$ be real numbers.

Let $\size {x - y}$ be the absolute value of $x - y$.

Then the function $d: \R^2 \to \R$:

$\map d {x, y} = \size {x - y}$

is called the distance between $x$ and $y$.

Complex Numbers

Let $x, y \in \C$ be complex numbers.

Let $\cmod {x - y}$ be the complex modulus of $x - y$.

Then the function $d: \C^2 \to \R$:

$\map d {x, y} = \cmod {x - y}$

is called the distance between $x$ and $y$.