Proving surjectivity in the following question.

Question

The question is given below:

My question is:

The onto part is not clear for me. Could anyone explain this for me please?

Answer 1

Let $\lambda \in D(V)$ be a linear functional on $V = \mathbb{Q}[X]$. Then $\lambda$ is determined by its action on the basis $\{1, X, X^2, \cdots\}$. Since $\delta_v(X^n) = n!b_{n+1}$, all you need is that $$ n!b_{n+1} = \lambda(X^n),$$ i.e. $b_{n+1} = \frac{1}{n!}\lambda(X^n)$, i.e. if $v = (\frac{1}{n!}\lambda(X^n))_{n \geq 0}$ then $\delta_v = \lambda$.