If in a orthogonal basis of a Hilbert space H, i take off just one element, do i get a strict subpsace of H?

Question

Let $B=\{b_n\}_{n \in \mathbb{N}}$ be a sequence of orthogonal elements of $\ell^2$ over $\mathbb{C}$ with $b_n \neq 0$

Let $A=B\setminus \{b_1\}$

I wolud like to know if is it true that:

$$ \overline{ \operatorname{span} A } \subsetneq \overline{ \operatorname{span} B } $$

Thanks.

Answer 1

As long as the $b_n$ are nonzero then yes, since $b_1$ is orthogonal to $\overline{\textrm{span}\,A}$.