In the event that the metric signature is used, Lorentzian -space is denoted ; the notation is used analogously with the metric signature .

The Lorentzian inner product induces a norm on Lorentzian space, whereby the squared norm of a vector has the form

(1)

Rewriting
(where
by definition), the norm in (0) can be written as

(2)

In particular, the norm induced by the Lorentzian inner product fails to be positive definite, whereby it makes sense to classify vectors in -dimensional Lorentzian space into types based on the sign
of their squared norm, e.g., as spacelike, timelike,
and lightlike. The collection of all lightlike vectors
in Lorentzian -space
is known as the light cone, which is further separated
into lightlike vectors which are positive and
negative lightlike. A similar distinction is
made for positive and negative
timelike vectors as well.

Sometimes, the -dimensional
Lorentzian norm is written to avoid confusion with the standard Euclidean
norm; one may also write for the Lorentzian inner product of two vectors and .

Lorentzian space comes up in a number of contexts throughout pure and applied mathematics. In particular, four-dimensional Lorentzian space is known as Minkowski
space and forms the basis of the study of spacetime within special relativity.
What's more, the collection

(3)

consisting of all vectors in having imaginary Lorentzian length forms a two-sheeted
hyperboloid of vectors satisfying the identity ; upon identifying antipodal
vectors of
(or, equivalently, upon discarding the negative sheet of vectors which satisfy
), one arrives at the so-called
hyperboloid model for hyperbolic -space .