Definition
The Lorentz space on a measure space
is the space of complex-valued measurable functions
on X such that the following quasinorm is finite

where
and
. Thus, when
,

and, when
,

It is also conventional to set
.
Decreasing rearrangements
The quasinorm is invariant under rearranging the values of the function
, essentially by definition. In particular, given a complex-valued measurable function
defined on a measure space,
, its decreasing rearrangement function,
can be defined as

where
is the so-called distribution function of
, given by

Here, for notational convenience,
is defined to be
.
The two functions
and
are equimeasurable, meaning that

where
is the Lebesgue measure on the real line. The related symmetric decreasing rearrangement function, which is also equimeasurable with
, would be defined on the real line by

Given these definitions, for
and
, the Lorentz quasinorms are given by

Lorentz sequence spaces
When
(the counting measure on
), the resulting Lorentz space is a sequence space. However, in this case it is convenient to use different notation.
Definition.
For
(or
in the complex case), let
denote the p-norm for
and
the ∞-norm. Denote by
the Banach space of all sequences with finite p-norm. Let
the Banach space of all sequences satisfying
, endowed with the ∞-norm. Denote by
the normed space of all sequences with only finitely many nonzero entries. These spaces all play a role in the definition of the Lorentz sequence spaces
below.
Let
be a sequence of positive real numbers satisfying
, and define the norm
. The Lorentz sequence space
is defined as the Banach space of all sequences where this norm is finite. Equivalently, we can define
as the completion of
under
.
Properties
The Lorentz spaces are genuinely generalisations of the
spaces in the sense that, for any
,
, which follows from Cavalieri's principle. Further,
coincides with weak
. They are quasi-Banach spaces (that is, quasi-normed spaces which are also complete) and are normable for
and
. When
,
is equipped with a norm, but it is not possible to define a norm equivalent to the quasinorm of
, the weak
space. As a concrete example that the triangle inequality fails in
, consider

whose
quasi-norm equals one, whereas the quasi-norm of their sum
equals four.
The space
is contained in
whenever
. The Lorentz spaces are real interpolation spaces between
and
.
Hölder's inequality
where
,
,
, and
.
Dual space
If
is a nonatomic σ-finite measure space, then
(i)
for
, or
;
(ii)
for
, or
;
(iii)
for
.
Here
for
,
for
, and
.
Atomic decomposition
The following are equivalent for
.
(i)
.
(ii)
where
has disjoint support, with measure
, on which
almost everywhere, and
.
(iii)
almost everywhere, where
and
.
(iv)
where
has disjoint support
, with nonzero measure, on which
almost everywhere,
are positive constants, and
.
(v)
almost everywhere, where
.