In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s,[1][2] are generalisations of the more familiar
The Lorentz spaces are denoted by
spaces, they are characterized by a norm (technically a quasinorm) that encodes information about the "size" of a function, just as the
The two basic qualitative notions of "size" of a function are: how tall is the graph of the function, and how spread out is it.
The Lorentz norms provide tighter control over both qualities than the
norms, by exponentially rescaling the measure in both the range (
norms, are invariant under arbitrary rearrangements of the values of a function.
is the space of complex-valued measurable functions
The quasinorm is invariant under rearranging the values of the function
In particular, given a complex-valued measurable function
defined on a measure space,
, its decreasing rearrangement function,
is the so-called distribution function of
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
However, in this case it is convenient to use different notation.
the Banach space of all sequences with finite p-norm.
the Banach space of all sequences satisfying
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
be a sequence of positive real numbers satisfying
The Lorentz sequence space
is defined as the Banach space of all sequences where this norm is finite.
The Lorentz spaces are genuinely generalisations of the
They are quasi-Banach spaces (that is, quasi-normed spaces which are also complete) and are normable for
is equipped with a norm, but it is not possible to define a norm equivalent to the quasinorm of
As a concrete example that the triangle inequality fails in
is a nonatomic σ-finite measure space, then (i)
has disjoint support, with measure