# Banach limit for Cesaro summable sequences

I'm solving an exercise from Lax's Functional analysis. The section concerns generalized limits (more particularly, Banach limits), which are obtained by applying the Hahn-Banach theorem to the classical limit functional on $$\ell^\infty$$.

The exercise wants me to construct a Banach limit which agrees with the Cesaro limit of Cesaro summable sequences, that is, bounded sequences such that "the arithmetic means of the partial sums converge."

If I understand this final sentence correctly (and wiki seems to confirm this), a sequence $$(x_1,x_2,\ldots)$$ is Cesaro summable if $$\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n s_k < \infty, \quad\text{where}\quad s_k = \sum_{j=1}^k x_j.$$ On the other hand, in this post, another definition of Cesaro summability is used, namely there it is required that $$\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n x_k < \infty.$$

My problem is that I would like to dominate the Cesaro limit functional by the (subadditive and positive homogeneous) function $$p(x) = \limsup_{n\to\infty} x_n,$$ which Lax uses to extend of the classical limit. This is easily done if we use the latter definition, but I'm not sure how to do it (or if it's even possible) if we use the former definition.

My other idea was to use $$p(x) = \limsup_{n\to\infty} |s_n|,$$ but this is not even well-defined since we are dealing with general bounded sequences.

How would you interpret the problem, i.e. which definition of Cesaro summability am I supposed to use? What is a good candidate for $$p$$ if we use the former definition?

• A sequence has a Cesàro limit if the sequence of averages converges. A series is Cesàro summable if the sequence of partial sums has a Cesàro limit. – Hagen von Eitzen Sep 23 '18 at 12:12
• I had seen this Terry Tao blog post on this matter: terrytao.wordpress.com/2017/05/11/… (but I don't know if it is useful or not) – Giuseppe Negro Sep 23 '18 at 12:13
• @HagenvonEitzen That's how I would interpret it too! But the exercise specifically says "[...] a sequence $(c_1, c_2, \ldots)$ that is Cesaro summable; namely the arithmetic means of the partial sums converge to $c$ [...]", which is what confused me. – MisterRiemann Sep 23 '18 at 12:15
• @GiuseppeNegro Thank you, I saw that one as well. I think that he also uses the second definition from my post, which seems like a reasonable thing to do. – MisterRiemann Sep 23 '18 at 12:16