Does the topology induced by a finite orthonormal basis have some special properties?

Question

Let $(H, \langle \cdot, \cdot \rangle)$ be a Hilbert space and $|\cdot|$ its induced norm. Let $(e_n)_{n=1}^N \subset H$. We define a map $d:H \times H \to \mathbb R$ by $$ d(x, y) := \sqrt{\sum_{n=1}^N \langle x-y, e_n \rangle^2} \quad \forall x,y \in H. $$

First, $d$ is a pseudo-metric on $H$. Second, the topology $\tau$ induced by $|\cdot|$ is finer than the topology $\tau'$ induced by $d$, i.e., $\tau' \subset \tau$.

We assume that $(e_n)_{n=1}^N$ is an orthonormal set. Does $\tau'$ have some special properties?

Answer 1

Denote by $K$ the subspace spanned by your orthonormal set. Your pseudo-metric is in fact the length of the orthogonal projection of $x-y$ onto $K.$ Is it helpful?

Anyway, to know something special or useful, you should have criteria.