= Banach limit
{c}
{title2=$L:\ell^\infty(\mathbb N;\mathbb R)\to\mathbb R$}
{wiki}
A Banach limit is a <positive linear functional> on the real <l-infinity sequence space> that extends the ordinary limit on the <convergent sequence space>, has <operator norm> one, and is invariant under the <left shift on bounded sequences>. The <Hahn-Banach theorem> constructs one by separating the constant sequence $\mathbf1$ from the subspace $(I-S)\ell^\infty$, whose distance from $\mathbf1$ is one. The functional is generally not unique.
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