Norm equivalence between inf-norm and polynomial coefficients

Question

Assume I have a normalised sequence of polynomials $(p_i)_{i = 0}^\infty$ ($\|p_i\|_\infty = 1$ for all $i = 0,1,\ldots$) such that $\mathcal{P}_n = \mathrm{span}\{p_0, \ldots, p_n\}$ for all $n = 0,1, \ldots$, and that $$ f(x) = \lim_{n \to \infty} \sum_{i = 0}^n c_i \, p_i(x). $$ It is straightforward to show $$ \|f\|_\infty \leq \|c\|_1. $$ Is it also possible to get a bound in the opposite direction, i.e. some result of the form $$ \|c\|_1 \leq C \, \|f\|_\infty? $$ I am particularly interested in the case of $p_i(x) = C \, x^i$ being the monomial basis, where here $C$ is the normalising factor to ensure $\|p_i\|_\infty = 1$. Also, I don't really need the 1-norm in the coefficient space, any other norm such that the space of infinite sequences in $\mathbb{C}$ is complete would do as well.