$\text { Given a sequence }\left\{x_{n}\right\}_{n \in \mathbb{N}} \text { in a Hilbert space } H,\text { for each } m \in \mathbb{N} \text { we have } x_{m} \notin \overline{\operatorname{span}}\left\{x_{n}\right\}_{n \neq m}, \text { i.e., } x_{m} \text { does not lie in the closed span of}\text { the other vectors (such a sequence is said to be minimal). }$
My question is does this 'minimal' fact implies all $x_{m} \notin \overline{\operatorname{span}}\left\{x_{n}\right\}_{n \neq m}$ orthogonal to $\overline{\operatorname{span}}\left\{x_{n}\right\}_{n \neq m}$?
No. For example, if $H$ is just $\mathbb{R}^\infty$, you could take the sequence $\{e_1, e_1 + e_2, e_3, e_4, e_5, \ldots\}$. The second vector is not orthogonal to the first but all the hypotheses are satisfied.