Omit an element of a Gram–Schmidt process

Question

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

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

We apply Gram–Schmidt process to the succession starting from $h_1$ then $h_2$ and so on obtaining the succession $\{u_n\}_{n \in \mathbb{N}}$

Let $W_n = \overline{\operatorname{span}}\{u_j\}_{j\ge n}$

My question is if $h_n \in W_{n+1}$

Thanks.

Answer 1

No. $h_n \in \operatorname{span}\{u_j\}_{j\le n}$, which is orthogonal to $W_{n+1}$.