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

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?

• What is your definition of $\Lambda$? (N.B. One really shouldn't speak of the Banach Limit.) – John Dawkins Mar 24 '16 at 0:48