The Lorentz spaces are denoted by . Like the spaces, they are characterized by a norm (technically a quasinorm) that encodes information about the "size" of a function, just as the norm does. 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 () and the domain (). The Lorentz norms, like the norms, are invariant under arbitrary rearrangements of the values of a function.
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
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 .
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.