L-infinity

In mathematics, ℓ^∞ and $L_{\infty }$ are two related vector spaces.

Sequence space

ℓ^∞ is a sequence space. Its elements are the bounded sequences. The vector space operations, addition and scalar multiplication, are applied coordinate by coordinate. ℓ^∞ is the largest ℓp space. With respect to the norm $\|x\|_{\infty }=\sup _{n}|x_{n}|,$ ℓ^∞ is also a Banach space.

Function space

$L_{\infty }$ is a function space. Its elements are the essentially bounded measurable functions. More precisely, $L_{\infty }$ is defined based on an underlying measure space, (S, Σ, μ). Start with the set of all measurable functions from S to R which are essentially bounded, i.e. bounded up to a set of measure zero. Two such functions are identified if they are equal almost everywhere. Denote the resulting set by L^∞(S, μ).

For a function  f  in this set, its essential supremum serves as an appropriate norm:

$\|f\|_{\infty }\equiv \inf\{C\geq 0:|f(x)|\leq C{\text{ for almost every }}x\}.$

See Lp space for more details.

Applications

One application of ℓ^∞ and $L_{\infty }$ is in economies with infinitely many commodities.^[1] In simple economic models, it is common to assume that there is only a finite number of different commodities, e.g. houses, fruits, cars, etc., so every bundle can be represented by a finite vector, and the consumption set is a vector space with a finite dimension. But in reality, the number of different commodities may be infinite. For example, a "house" is not a single commodity type since the value of a house depends on its location. So the number of different commodities is the number of different locations, which may be considered infinite. In this case, the consumption set is naturally represented by $L_{\infty }$.

References

1. Bewley, T. F. (1972). "Existence of equilibria in economies with infinitely many commodities". Journal of Economic Theory. 4 (3): 514. doi:10.1016/0022-0531(72)90136-6.