Dual of the polynomials

The dual of the space of polynomials $R =\mathbbm{K}[x_1, \ldots, x_n]$ is the set $\hat{R}$ of linear forms $\Lambda : \mathbbm{K}[x_1, \ldots, x_n] \rightarrow \mathbbm{K}$. An element $\Lambda$ is represented by a formal power series

\[ \Lambda = \sum_{\alpha} \Lambda (\mathbf{x}^{\alpha})\mathbf{d}^{\alpha} = \sum_{\alpha} \Lambda_{\alpha} \mathbf{d}^{\alpha}, \]

where $(\mathbf{d}^{\alpha})$ is the dual basis of the monomial basis $(\mathbf{x}^{\alpha})$. The set of linear forms is a $R$-module:

\[ p \cdot \Lambda (q) = \Lambda (p q) \]

In particular, we have $x_i \cdot d_1^{\alpha_1} \cdots d_n^{\alpha_n} = d_1^{\alpha_1} \cdots d_{i - 1}^{\alpha_{i - 1}} d_i^{\alpha_i - 1} d_{i + 1}^{\alpha_{i + 1}} \cdots d_n^{\alpha_n}$ if $\alpha_i > 0$ and $0$ otherwise.