Problem
Let $V$ be the vector space of all polynomial real coefficients and let $W$ be the subspace of $V^{\ast}$(The dual of $V$) spanned by the linear forms $(l_n)_{n\ge 0}$, where $l_n(P)=P^n(0)$. Prove that $W^{\perp}=${0},but $W^{\perp}\ne V$
The first part was quite easy and comes from definition of $W^{\perp}$. But I'm stuck at the inequality part. The objective is to find a linear form which is not linear combination of $l_n's$ .
Here I'm seeing for any polynomial $P(x)=a_0+a_1x+...+a_nx^n$ the coefficients can be written as $a_n=\frac{l_n(P)}{n!}$ .Then any new linear form created by using only coefficients will result the new linear form is linear combination of $l_n$ 's. Please provide some hints to how to create a different linear form other than $l_n$'s.