${L}^{\mathrm{\infty}}(X,\mu )$
Let $X$ be a nonempty set and $\mathcal{A}$ be a $\sigma $algebra on $X$. Also, let $\mu $ be a nonnegative measure^{} defined on $\mathcal{A}$. Two complex valued functions $f$ and $g$ are said to be equal almost everywhere on $X$ (denoted as $f=g$ a.e. if $\mu \{x\in X:f(x)\ne g(x)\}=0.$ The relation^{} of being equal almost everywhere on $X$ defines an equivalence relation^{}. It is a common practice in the integration theory to denote the equivalence class^{} containing $f$ by $f$ itself. It is easy to see that if ${f}_{1},{f}_{2}$ are equivalent^{} and ${g}_{1},{g}_{2}$ are equivalent, then ${f}_{1}+{g}_{1},{f}_{2}+{g}_{2}$ are equivalent, and ${f}_{1}{g}_{1},{f}_{2}{g}_{2}$ are equivalent. This naturally defines addition and multiplication among the equivalent classes of such functions. For a measureable $f:X\to \u2102$, we define
$${\parallel f\parallel}_{\text{ess}}=\mathrm{inf}\{M>0:\mu \{x:f(x)>M\}=0\},$$ 
called the essential supremum^{} of $f$ on $X$. Now we define,
$$ 
Here the elements of ${L}^{\mathrm{\infty}}(X,\mu )$ are equivalence classes.
Properties of ${L}^{\mathrm{\infty}}(X,\mu )$

1.
The space ${L}^{\mathrm{\infty}}(X,\mu )$ is a normed linear space with the norm ${\parallel \cdot \parallel}_{\text{ess}}$. Also, the metric defined by the norm is complete^{}, making ${L}^{\mathrm{\infty}}(X,\mu )$, a Banach space^{}.

2.
${L}^{\mathrm{\infty}}(X,\mu )$ is the dual of ${L}^{1}(X,\mu )$ if $X$ is $\sigma $finite.

3.
${L}^{\mathrm{\infty}}(X,\mu )$ is closed under pointwise multiplication, and with this multiplication it becomes an algebra. Further, ${L}^{\mathrm{\infty}}(X,\mu )$ is also a ${C}^{*}$algebra (http://planetmath.org/CAlgebra) with the involution defined by ${f}^{*}(x)=\overline{f(x)}$. Since this ${C}^{*}$algebra is also a dual of some Banach space, it is called von Neumann algebra^{}.
Title  ${L}^{\mathrm{\infty}}(X,\mu )$ 

Canonical name  LinftyXmu 
Date of creation  20130322 13:59:46 
Last modified on  20130322 13:59:46 
Owner  ack (3732) 
Last modified by  ack (3732) 
Numerical id  11 
Author  ack (3732) 
Entry type  Definition 
Classification  msc 28A25 