Convergence of a succession obtained by the Gram–Schmidt process

Question

Let $H$ be an Hilbert space over $\mathbb{C}$.

Let $\{h_n\}_{n \in \mathbb{N}} \subset H$ be a sequence of linearly independent vectors in $H$ such that $h_n \to h \neq 0$ in norm topology.

We apply Gram–Schmidt process, without normalizing, to $\{h_n\}_{n \in \mathbb{N}} \cup \{h\}$ starting from $h$ and then $h_1, h_2,\dots$ obtaining the sequence $\{v_n\}_{n \in \mathbb{N}}$.

My question is if is it true that $v_n \to 0$ in norm topology or that $v_n \overset{w}{\to} 0$ in weak topology.

Thank you for all suggestions.