An Orthonormal Basis of the Ambient Space Contained in the Form Space of a Hilbert Triple

In the setting of Hilbert Triples: Standing Notation and Background, let $(H,V,A)$ be the Hilbert triple fixed there, so that $V$ is a linear subspace of $H$ that is dense in $H$ and satisfies $|x|_{H}\le|x|_{V}$ for every $x\in V$, and the metric space $(V,d_{V})$ is separable. Assume that $H$, as a vector space over $\mathbb{R}$, is not finite-dimensional.

(i) Then there is an orthonormal basis $(e_{k})_{k\in\mathbb{N}}$ of $H$ such that $e_{k}\in V$ for every $k\in\mathbb{N}$.