= Cesàro construction of a Banach limit
{c}
{title2=$p(x)=\limsup_N N^{-1}\sum_{j=1}^N x_j$}
The displayed <sublinear functional> on the real <l-infinity sequence space> agrees with the ordinary limit on <convergent sequences>. A dominated extension by the <Hahn-Banach theorem> therefore defines a <generalized limit> bounded between the lower and upper limits of the <Cesàro means>. For the <left shift on bounded sequences> $S$, the averages of $Sx-x$ equal $(x_{N+1}-x_1)/N$, which tend to zero. Domination applied to both signs gives $L(Sx-x)=0$, hence shift invariance. Positivity, agreement on constant sequences and the <operator norm> one follow from the bounds, so $L$ is a <Banach limit>.
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