Let $X$ be a Banach space and $H_b (X)$ be the algebra of complex-valued entire functions on $X$ which are bounded on bounded sets, with the topology of uniform convergence on bounded sets.
Let $\varphi \in H_b^* (X)$ (which is the dual space of $H_b (X)$).
Each $\varphi \in H_b^* (X)$ is continuous with respect to the norm of uniform convergence on some ball in $X$.
Define the radius function $R$ on $H_b^* (X)$ by declaring $R(\varphi) $ to be the infimum of all $r >0$ such that $\varphi$ is continuous with respect to the norm of uniform convergence on the ball $r B$.
I don't understand the above bold statement. I am a bit confused because I think $\varphi \in H_b^* (X)$ is continuous on every ball in $X$ (so, the radius of $\varphi$ is always $0$).
I want to know what I missed.