I'm reading Costello's book "Renormalization and effective field theories" (preliminary PDF, p. 35). I am stuck on his discussion of Feynman diagrams.
He considers a finite-dimensional super vector space $U$ and the completed symmetric algebra $\mathcal{O}(U)$ over its dual, namely $$ \bar{S}(U^*) = \prod_{p=0}^{\infty} \mathrm{Sym}^p(U^*) $$ and he says that it is the ring of power series in a variable $u \in U$. How does that identification work? In what ring do the coefficients of the power series lie?
Then, for a homogeneous element $f \in \mathcal{O}(U)$ of degree $k$, he defines the map $$ D^k f : U^{\otimes k} \to \mathbb{K}, \quad D^k f(u_1 \otimes \dots \otimes u_k) = (\frac{\partial}{\partial u_1} \dots \frac{\partial}{\partial u_k} f)(0). $$ What do those partial derivatives mean? Are they just formal differentiation of the power series followed by evaluation? If so, how does the evaluation work?
