In mathematics, ℓ ∞ {\displaystyle \ell ^{\infty }} , the (real or complex) vector space of bounded sequences with the supremum norm, and L ∞ = L ∞ ( X , Σ , μ ) {\displaystyle L^{\infty }=L^{\infty }(X,\Sigma ,\mu )} , the vector space of essentially bounded measurable functions with the essential supremum norm, are two closely related Banach spaces. In fact the former is a special case of the latter. As a Banach space they are, respectively, 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 (if the measure space fulfills the conditions of being localizable and therefore semifinite). Pointwise multiplication gives them the structure of a Banach algebra, and in fact they are the standard examples of abelian Von Neumann algebras.
Sequence space 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} . This space is the strong dual space of ℓ 1 {\displaystyle \ell ^{1}} : indeed, every x ∈ ℓ ∞ {\displaystyle x\in \ell ^{\infty }} defines a continuous functional on the space ℓ 1 {\displaystyle \ell ^{1}} of absolutely summable sequences by component-wise multiplication and summing:
ℓ ∞ → ( ℓ 1 ) ′ {\displaystyle {\begin{aligned}\ell ^{\infty }&\to ({\ell ^{1}})'\end{aligned}}}
via
x ↦ ( y ↦ ∑ i = 1 ∞ x i y i ) {\displaystyle {\begin{aligned}x&\mapsto \left(y\mapsto \sum _{i=1}^{\infty }x_{i}y_{i}\right)\end{aligned}}}
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.
( ℓ 1 ) ′ = ℓ ∞ {\displaystyle ({\ell ^{1}})'=\ell ^{\infty }}
However, not every continuous linear functional on ℓ ∞ {\displaystyle \ell ^{\infty }} arises from an absolutely summable series in ℓ 1 , {\displaystyle \ell ^{1},} and hence ℓ ∞ {\displaystyle \ell ^{\infty }} is not a reflexive Banach space.
Function space
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