L-infinity

In mathematics, ℓ ∞ {displaystyle ell ^{infty }} , the (real or complex) vector space of bounded sequences, and L ∞ = L ∞ ( X , Σ , μ ) {displaystyle L^{infty }=L^{infty }(X,Sigma ,mu )} , the vector space of essentially bounded measurable functions, are two closely related Banach spaces with respect to the natural norm ‖ − ‖ ∞ {displaystyle |-|_{infty }} , the supremum respectively essential supremum norm. In fact the former is a special case of the latter. As a Banach space they are the continuous dual of the Banach spaces ℓ 1 {displaystyle ell _{1}} of absolutely summable sequences, and L 1 = L 1 ( X , Σ , μ ) {displaystyle L^{1}=L^{1}(X,Sigma ,mu )} of absolutely integrable measurable functions. Point wise multiplication gives them the structure of a Banach algebra, in fact they are the standard examples of abelian Von Neumann algebras. In mathematics, ℓ ∞ {displaystyle ell ^{infty }} , the (real or complex) vector space of bounded sequences, and L ∞ = L ∞ ( X , Σ , μ ) {displaystyle L^{infty }=L^{infty }(X,Sigma ,mu )} , the vector space of essentially bounded measurable functions, are two closely related Banach spaces with respect to the natural norm ‖ − ‖ ∞ {displaystyle |-|_{infty }} , the supremum respectively essential supremum norm. In fact the former is a special case of the latter. As a Banach space they are the continuous dual of the Banach spaces ℓ 1 {displaystyle ell _{1}} of absolutely summable sequences, and L 1 = L 1 ( X , Σ , μ ) {displaystyle L^{1}=L^{1}(X,Sigma ,mu )} of absolutely integrable measurable functions. Point wise multiplication gives them the structure of a Banach algebra, in fact they are the standard examples of abelian Von Neumann algebras. The vector space ℓ ∞ {displaystyle ell ^{infty }} is a sequence space whose elements are the bounded sequences. The vector space operations, addition and scalar multiplication, are applied coordinate by coordinate. With respect to the norm ‖ x ‖ ∞ = sup n | x n | {displaystyle |x|_{infty }=sup _{n}|x_{n}|} , ℓ ∞ {displaystyle ell ^{infty }} is a standard example of a Banach space. In fact, ℓ ∞ {displaystyle ell ^{infty }} can be considered as the ℓ p {displaystyle ell ^{p}} space with the largest p {displaystyle p} . Moreover, every x ∈ ℓ ∞ {displaystyle xin ell ^{infty }} defines a continuous functional on the space ℓ 1 {displaystyle ell ^{1}} of absolutely summable sequences by point wise multiplication and summing: By evaluating on ( 0 , … , 0 , 1 , 0 , … ) {displaystyle (0,ldots ,0,1,0,ldots )} we see that every continuous linear functional on ℓ 1 {displaystyle ell ^{1}} arises in this way. i.e. Not every continuous linear functional on ℓ ∞ {displaystyle ell ^{infty }} arises from a absolutely summable series however: ℓ 1 {displaystyle ell ^{1}} , and hence l ∞ {displaystyle l^{infty }} is not a reflexive Banach space. L∞ is a function space. Its elements are the essentially bounded measurable functions. More precisely, L∞ 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: See Lp space for more details. The sequence space is a special case of the function space: l ∞ = L ∞ ( N ) {displaystyle l_{infty }=L_{infty }(mathbb {N} )} where the natural numbers are equipped with the counting measure. One application of ℓ∞ and L∞ is in economies with infinitely many commodities. 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∞.