  Let $H$ be a Hilbert space that can be embedded as a dense subspace of a
Banach space $X$ such that the norm of the embedding is equal to $1$. We
consider the following statements for a nonzero vector $\varphi$ in $H$:
  (A) $\|\varphi\|_X = \|\varphi\|_H$.
  (H) $\|\varphi+f\|_X \geq \|\varphi\|_X$ for every $f$ in $H$ such that
$\langle f, \varphi \rangle =0$. We use duality arguments to establish that (A)
$\implies$ (H), before turning our attention to the special case when the
Hilbert space in question is the Hardy space $H^2(\mathbb{T}^d)$ and the Banach
space is either the Hardy space $H^1(\mathbb{T}^d)$ or the weak product space
$H^2(\mathbb{T}^d) \odot H^2(\mathbb{T}^d)$. If $d=1$, then the two Banach
spaces are equal and it is known that (H) $\implies$ (A). If $d\geq2$, then the
Banach spaces do not coincide and a case study of the polynomials
$\varphi_\alpha(z) = z_1^2 + \alpha z_1 z_2 + z_2^2$ for $\alpha\geq0$
illustrates that the statements (A) and (H) for the two Banach spaces describe
four distinct sets of functions.
