The notion of a Banach limit is usually defined for the space of bounded sequences, but one can define it for more general spaces, see (see "What is a generalized limit?, " and "Do multiplicative Banach limits exist?" and references therein).
I am interested in the Banach space of Bounded functions $f: \mathbb{R}_+ \to \mathbb{C}$ with $ ||f|| = \rm sup_{t \in \mathbb{R}_+} |f(t)| $respect to the uniform norm $ \|f\| = \sup_{t \in \mathbb{R}_+} |f(t)|$ and Banach limits on it. Consider a sequence $f_n(t)$ ,$n\in \mathbb{N}$, of such functions and a Banach limit denoted by $L$. Suppose that for all $t\in \mathbb{R}_+$: $\sum_{n=1}^\infty f_n(t)$ exists, and consider its Banach limit $$L\big( \sum_{n=1}^{\infty} f_n(t) \big) $$$$L\Big( \sum_{n=1}^{\infty} f_n(t) \Big) $$ Is it true that $$L\big( \sum_{n=1}^{\infty} f_n(t) \big) = \sum_{n=1}^{\infty}L\big( f_n(t) \big) $$$$L\Big( \sum_{n=1}^{\infty} f_n(t) \Big) = \sum_{n=1}^{\infty}L\big( f_n(t) \big) $$
for all Banach limits $L$?
With usual limits, i.e.\ $\lim_{t\to \infty} \lim_N \sum_{n=1}^N f_n(t) $, it's not always the case that we can commute the two limits.
So, does the dominated convergence theorem extend to Banach limits, so that the Banach limit $L$ can be moved inside the summation/commute the limits? Under what assumptions?
I'd appreciate references.
