I asked this question on math.SE, but couldn't get an answer.
Let $H^\infty(\mathbb{D})$ denote the set of functions holomorphic and bounded on $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$. Conseqently, $H^\infty(\mathbb{D}^n)$ denotes the set of bounded holomorphic functions on the polydisk $\mathbb{D}^n = \mathbb{D} \times \ldots \times \mathbb{D}$.
With the norm $\|f\|_{H^\infty(\mathbb{D}^n)} := \sup_{z \in \mathbb{D}^n} |f(z)|$ these sets become Banach spaces (i.e. Hardy-space with $p=\infty$).
Is it true, that $$H^\infty(\mathbb{D}) \otimes H^\infty(\mathbb{D}) = H^\infty(\mathbb{D}^2),$$ where $\otimes$ denotes the completion with respect to the $H^\infty(\mathbb{D}^2)$-norm?
