Consider the uncountable-dimensional vector space $V$ consisting of finite linear combinations of infinite sequences of the form $(1,z,z^2,z^3,\dots)$ with $z \neq 1$ in $\mathbb{C}$. Since the sequences $(1,z,z^2,z^3,\dots)$ are linearly independent, there is a linear function $\sigma$ from $V$ to $\mathbb{C}$ that sends the sequence $(1,z,z^2,z^3,\dots)$ to the number $1/(1-z)$; for example, it sends $(2^0+(\frac12)^0,2^1+(\frac12)^1,2^2+(\frac12)^2,\dots)$ to $(1/(1-2))+(1/(1-\frac12)) = -1 + 2 = 1$; thus, in $V$, we could say that $2+\frac{5}{2}+\frac{17}{4}+\dots=1$. (This example sequence was designed to thwart certain approaches to regularizing divergent series, by way of combining two series that converge with respect to different Ostrowski valuations.)

$V$ would be a natural setting in which to do certain kinds of Euler-ish calculations with divergent series, but unfortunately $V$ lacks some desirable closure properties; for instance, it doesn't contain convolutions like $(1,w+z,w^2+wz+z^2,w^3+w^2z+wz^2+z^3,\dots$) which arise when you multiply two geometric series.

Is there an extension of $V$ to a larger space $V'$ that is closed under both addition and convolution, and an extension of $\sigma$ to a linear map from $V'$ to $\mathbb{C}$? (Is it described in the existing literature on divergent series?)