\begin{definition}[Definition:Banach Limit]
Let $\struct {\map {\ell^\infty} \R, \norm \cdot_\infty}$ be the normed vector space of bounded sequences on $\R$. 
Let $\struct {\paren {\map {\ell^\infty} \R}^\ast, \norm \cdot_{\paren {\ell^\infty}^\ast} }$ be the normed dual space of $\struct {\map {\ell^\infty} \R, \norm \cdot_\infty}$.
Let $S : \map {\ell^\infty} \R \to \map {\ell^\infty} \R$ be the left shift operator on $\map {\ell^\infty} \R$. 
Let $\mathbf 1$ be the sequence with ${\mathbf 1}_n = 1$ for each $n \in \N$. 
We say that $L \in \paren {\map {\ell^\infty} \R}^\ast$ is a '''Banach limit''' {{iff}}: 
:$(1): \quad$ $\map L x \ge 0$ for all $x = \sequence {x_n}_{n \mathop \in \N} \in \map {\ell^\infty} \R$ with $x_n \ge 0$ for each $n \in \N$
:$(2): \quad$ $\map L x = \map L {S x}$ for all $x \in \map {\ell^\infty} \R$
:$(3): \quad$ $\map L {\mathbf 1} = 1$.
\end{definition}