Norm attaining vectors and Hilbert points

Abstract: 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.