Multiplying Balls in Cⁿ[0,1]

Let $C^{(n)}[0,1]$ stand for the Banach space of functions $f:[0,1]\rightarrow \mathbb{R}$ with continuous $n$ -th derivative. We prove that if $B_{1},B_{2}$ are open balls in $C^{(n)}[0,1]$ then the set $B_{1}\cdot B_{2}=\{f\cdot g:f\in B_{1},g\in B_{2}\}$ has non-empty interior in $C^{(n)}[0,1].$ This extends the result of [1] dealing with the space of continuous functions on $[0,1]$.