On the first page of Chapter 1 of Rieffel's *Deformation Quantization for Actions of $ \mathbb{R}^{d} $*, Rieffel defines a family of seminorms on the space $ A^{\infty} $ of smooth vectors of a Fréchet space $ A $, for some action $ \alpha $ of the Lie group $ \mathbb{R}^{d} $ on $ A $, as follows. Suppose we already have a family $ (\| \cdot \|_{i})_{i \in \mathbb{N}} $ of seminorms on $ A $ that determine its topology. Choose a basis $ \lbrace X_{1},\ldots,X_{d} \rbrace $ of $ \mathbb{R}^{d} $. Then for each $ k \in \lbrace 1,\ldots,d \rbrace $, let $ \alpha_{X_{k}} $ denote the operator of partial differentiation on $ A^{\infty} $ in the direction of $ X_{k} $; we thus identify $ \mathbb{R}^{d} $ with its Lie algebra in the usual way. For convenience, denote $ \alpha_{X_{k}} $ simply by $ \partial_{k} $. Next, for any multi-index $ \mu = (\mu_{1},\ldots,\mu_{d}) \in \mathbb{N}_{0}^{d} $, let $ \partial^{\mu} $ denote the higher-order partial derivative $ \partial_{1}^{\mu_{1}} \cdots \partial_{d}^{\mu_{d}} $. Then equip $ A^{\infty} $ with the seminorms
$$
\forall (j,k) \in \mathbb{N} \times \lbrace 1,\ldots,d \rbrace ~~ \& ~~ \forall a \in A^{\infty}: \quad \| a \|_{j,k} \stackrel{\text{def}}{=} \sup_{1 \leq i \leq j} \sum_{|\mu| \leq k} \frac{\| \partial^{\mu} a \|_{i}}{\mu!},
$$
where $ |\mu| \stackrel{\text{def}}{=} \mu_{1} + \cdots + \mu_{d} $ and $ \mu! \stackrel{\text{def}}{=} \mu_{1}! \cdots \mu_{d}! $.

My question is: As we are applying partial derivatives to $ a \in A^{\infty} $, are we identifying $ a $ with the function $ f_{a}: \mathbb{R}^{d} \rightarrow A $ defined by $ {f_{a}}(\mathbf{x}) \stackrel{\text{def}}{=} \alpha(\mathbf{x},a) $?

Thank you very much in advance!