Definition of Banach Limits on $\ell^\infty$. Proof of Linearity and Continuity

Question

I want to show that the Banach Limit $\Lambda$ on the set $\ell^\infty$ is a continuous linear functional in the dual space $(\ell^\infty)^\star$. I know that the Banach Limit exists, is left-translation invariant and sitting in between $\lim \inf x_n$ and $\lim \sup x_n$ for any sequence $x = (x_n)$ from $\ell^\infty$. How can I show linearity directly using the definition here given? And for continuity I guess I will have to show $\Lambda$ is bounded in a nbhd of the $0$-sequence and non-zero?

Answer 1

Your definition has two properties: $lim inf < Lim < lim sup$ and $Lim(x)=Lim(Sx)$

The average of the $lim inf$ and $lim sup$ is a non-linear functional that satisfies the two given properties (and even extends the classical limit and has positivity). Take the absolutely almost convergent sequence $(0,1,1,0,1,1,0,\dots)$ with limit $2/3$ for any Banach limit. Its $lim sup$ is $1$, and its $liminf$ is $0$. Their average, $1/2$, isn’t $2/3$. Seeing as this operator has every property of the Banach limit besides linearity, but disagrees with every Banach limit, it can be concluded that linearity is the missing property.