one considers the Banach-$*$-Algebra (involution=conjugation, multiplication = pointwise multiplication):
$$A_r:=\{f:\mathbb{R}^n \to \mathbb{C} \mid f=\sum\limits_{m=0}^\infty f_m, \\ \|f\|_r < \infty\},$$
where the f_m are polynomials homgoeneous of degree m and $$\|f\|_r =\sum\limits_{m=0}^\infty \|f_m\|_{\infty, \overline{B_r(0)}}$$ the series over the maximal norm of the f_n on the closed ball with radius r and center 0.
I would like to know, if somebody knows a description of the dual of this Banach space?
Best regards, Dominik
Your space is just the $\ell^1$ direct sum of the spaces $E_n$ formed by the homogeneous polynomials of degree $n$. The dual is then the $\ell^\infty$ direct sum of the duals of the $E_n$.