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:

- $\mathcal{S} \subset \mathbb{R}^\mathbb{N}$ is a vector subspace which contains the space $\mathcal{C}$, of real sequences whose sum converges, and is closed under $\shft$.
- $\sigma \colon \mathcal{S} \to \mathbb{R}$ is a linear operator.
- Regular: For every $(a_n)_n \in \mathcal{C}$ we have that $\sigma ( (a_n)_n ) =\Sigma_n a_n=a_1+a_2+\cdots$.
- 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).

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