L-infinity

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In mathematics, ${\displaystyle \ell ^{\infty }}$, the (real or complex) vector space of bounded sequences with the supremum norm, and ${\displaystyle L^{\infty }=L^{\infty }(X,\Sigma ,\mu )}$, 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 the continuous dual of the Banach spaces ${\displaystyle \ell _{1}}$ of absolutely summable sequences, and ${\displaystyle L^{1}=L^{1}(X,\Sigma ,\mu )}$ of absolutely integrable measurable functions (if the measure space fulfills the conditions of being localizable and therefore semifinite).^[1] Pointwise multiplication gives them the structure of a Banach algebra, and in fact they are the standard examples of abelian Von Neumann algebras.