Let $V = \mathbb{Q}[X]$ and to each vector $v = [b_1, b_2, . . .] \in \mathbb{Q}^{\infty}$ assign a linear functional $\delta_v \in D(V)$ defined by $\delta_v: \sum_{n=0}^{\infty}a_nX_n \mapsto \sum_{n=0}^{\infty} n! a_n b_{n+1}$. Is the function $\alpha : \mathbb{Q}^{\infty} \to D(V)$ defined by $v \mapsto \delta_v$ an isomorphism?
I suspect that I will have to show that $\alpha$ is both monic (one-to-one) and epic (onto) to show that it is bijective, but I am not sure how to accomplish that.
Any ideas/suggestions are greatly appreciated.