If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift operator of summation. These, of course, are not real proofs, since the series do not converge, but one might try to generalize the concept of summation to such series. $\DeclareMathOperator{\shft}{sh}$


Thus, inspired by Lebesgue measure on the reals, one might define a summability space as a pair $(\mathcal{S}, \sigma)$ such that:


1. $\mathcal{S} \subset \mathbb{R}^\mathbb{N}$ is a vector subspace which contains the space $\mathcal{C}$, of real sequences whose sum converges, and is closed under $\shft$.
2. $\sigma \colon \mathcal{S} \to \mathbb{R}$ is a linear operator.
3. Regular: For every $(a\_n)\_n \in \mathcal{C}$ we have that
$\sigma ( (a\_n)\_n ) =\Sigma\_n a\_n=a\_1+a\_2+\cdots$.
4. Translative: For every $(a\_n)\_n \in \mathcal{S}$ we have that $\sigma((a\_n)\_n)=a\_1 +\sigma(\shft((a\_n)\_n))$.


Here $\shft$ is the shift operator, i.e., $\shft(a\_1, a\_2, \dots)=(a\_2,a\_3,\dots)$.


What is the largest possible summability space (which is nice is some way)? Has this idea already been studied? Can we strengthen the definition of a summability space to get a canonical largest summability space (in the same way that Lebesgue measure is the "largest nice" measure on $\mathbb{R}$)?


I know that there are many ways to sum divergent series, such as Cesaro summation, Abel summation, etc. But I am asking about the "best" summation method, which "unifies" all other summation method. (Note that we can't simply apply the Hahn-Banach theorem or such a result since we would like to preserve translativity as well).



Generalizations of summation methods of divergence series
$\newcommand{\R}{\mathbb R}\newcommand{\N}{\mathbb N}\newcommand{\si}{\sigma}\newcommand{\SSS}{\mathcal S}\newcommand{\CC}{\mathcal C}\newcommand{\sh}{\operatorname{sh}}$First of all, as was noted in the previous [comment](https://mathoverflow.net/questions/432344/generalizations-of-summation-methods-of-divergence-series?noredirect=1#comment1112828_432344), $\CC$ should be defined, not as the set of all convergent sequences in $\R^\N$, but as the set of all sequences in $\R^\N$ summable (say) in the sense that the corresponding sequence of partial sums is convergent.


Let now $T:=\sh$ and $\N\_0:=\{0\}\cup\N$.


Partially ordering the summability spaces by inclusion and using Zorn's lemma, we see that there is a maximal summability space.


Actually, there are infinitely many maximal summability spaces.


Indeed, take any $t\in\R$. Let
\begin{equation\*}
 \SSS\_t:=\text{span}(\CC\cup\{T^k b\colon k\in\N\_0\}), \tag{1}\label{1}
\end{equation\*}
where
\begin{equation\*}
 b:=(b\_1,b\_2,\dots)\quad\text{with}\quad b\_n:=n!. 
\end{equation\*}


Note that, for each nonzero sequence $a\in\CC$, the sequences $a,b,Tb,T^2b,\dots$ are linearly independent. To see why this is true, suppose that, to the contrary, $c\_{-1} a+c\_0 b+c\_1 Tb+\dots+c\_k T^kb=0$
for some $k\in\N\_0$ and some real $c\_{-1},\dots,c\_k$ such that $c\_k\ne0$. Then for all $n\in\N$ we have
\begin{equation}
c\_{-1} a\_n+c\_0 n!+c\_1(n+1)!+\dots+c\_{k-1}(n+k-1)!+c\_k(n+k)!=0. 
\end{equation}
Letting $n\to\infty$, we get
\begin{equation}
c\_{-1} a\_n+c\_0 n!+c\_1(n+1)!+\dots+c\_{k-1}(n+k-1)!=o((n+k)!) 
\end{equation}
and hence $c\_k=0$, a contradiction.


So, we have the summability space $(\SSS\_t,\si\_t)$ with $\SSS\_t$ as in \eqref{1} and the linear functional $\si\_t\colon\SSS\_t\to\R$ defined by the conditions
\begin{equation\*}
 \si\_t(a):=\sum\_{n=1}^\infty a\_n
\end{equation\*}
for $a=(a\_1,a\_2,\dots)\in\CC$ and  

\begin{equation\*}
 \si\_t(T^k b):=t-\sum\_{n=1}^k b\_n 
\end{equation\*}
for $k\in\N\_0$ (so that $\si\_t(b)=t$).
**(Added detail:** Indeed, in view of \eqref{1} and because the sequences $a,b,Tb,T^2b,\dots$ are linearly independent for any nonzero $a\in\CC$, each sequence $s\in\SSS\_t$ can be *uniquely* represented by the formula $s=a+c\_0 b+c\_1 Tb+\dots+c\_k T^kb$ for some $a\in\CC$, some $k\in\N\_0$, and some real $c\_0,\dots,c\_k$; and then $\si\_t(s)=\si\_t(a)+c\_0 \si\_t(b)+c\_1 \si\_t(Tb)+\dots+c\_k \si\_t(T^kb)$ by the linearity of $\si\_t$.**)**


Clearly then, no summability space can contain two summability spaces of the form $(\SSS\_t,\si\_t)$ with two different values of $t\in\R$. On the other hand, by Zorn's lemma, each summability space of the form $(\SSS\_t,\si\_t)$ is contained in a maximal summability space $(\SSS^\*\_t,\si^\*\_t)$, and all these maximal summability spaces $(\SSS^\*\_t,\si^\*\_t)$ are distinct from one another.


Thus, as claimed, there are infinitely many maximal summability spaces and, therefore, as stated in the previous [comment](https://mathoverflow.net/questions/432344/generalizations-of-summation-methods-of-divergence-series?noredirect=1#comment1112826_432344), **there is no one largest summability space**.


(This is probably why apparently ["nobody studied"](https://mathoverflow.net/questions/432344/generalizations-of-summation-methods-of-divergence-series?noredirect=1#comment1112872_432344) such a nonexistent entity.)


