TITLE: Continuity of Banach limit and existence of $\Lambda_0\in(\ell^\infty/c_0)^*$ such that $\Lambda=\Lambda_0\circ q_0$, with $q_0$ the quotient map
QUESTION [5 upvotes]: Let $\Lambda$ be any Banach limit on $\ell^\infty$, where $\ell^\infty$ denotes the space of bounded real sequences. A Banach limit is defined as a linear functional $\Lambda$ such that
$$ \Lambda(\tau x)=\Lambda(x), \forall x\in\ell^\infty$$
$$ \liminf_{n\rightarrow\infty}x_n\leq\Lambda(x)\leq\limsup_{n\rightarrow\infty}x_n$$
where we write $x=(x_n)_{n\in\mathbb{N}}$ for a sequence $x\in\ell^\infty$ and we define left translation on $\ell^\infty$ by $(\tau x)_n=x_{n+1},n=1,2,\dots$.
I would like to show that $\Lambda\in(\ell^\infty)^*$, which means that $\Lambda$ is a continuous, linear functional on $\ell^\infty$. Thus I need to show that $\Lambda$ is continuous. How do I do this?
Furthermore I wish to show that there exists a continuous, linear functional $\Lambda_0\in(\ell^\infty/c_0)^*$ such that $\Lambda=\Lambda_0\circ q_0$, where $$q_0:\ell^\infty\rightarrow\ell^\infty/c_0$$ is the quotient map and
$$c_0=\{(x_n)\in\ell^\infty\mid \lim_{n\rightarrow\infty}x_n=0\}$$
I can't seem to get anywhere with these questions. Any help is greatly appreciated.

REPLY [2 votes]: That $\Lambda$ is continuous follows directly from the second estimate:
$$
-\|x\|_\infty\leq \liminf_{n\to\infty} \Lambda(x_n)\leq \Lambda(x)\leq \limsup_{n\to\infty}\Lambda(x_n)\leq\|x\|_\infty
$$
Thus, $|\Lambda(x)|\leq \|x\|_\infty$ for all $x\in \ell^\infty$.
For your second claim define $\Lambda_0(x+c_0):=\Lambda(x)$. This is well-defined since $x-y\in c_0$ implies $\Lambda(x-y)=0$, i.e. $\Lambda(x)=\Lambda(y)$.