Orthonormalize subsets in a sequence of subsets of a Hilbert space

Question

Let $H$ be a infinite dimensional, separable hilbert space over the complex plane $\mathbb{C}$

Let $\{A_n\}_{n \in \mathbb{N}} \in H$ be a sequence of subsets of $H$ such that $\forall n \in \mathbb{N}:$

$A_n$ is linearly independent, infinite, countable.

Let $A= \bigcup_{n=1}^\infty A_n$ and $A$ is linearly independent.

Is it possibile orthonormalize $A_n$ getting $B_n$ such that $B= \bigcup_{n=1}^\infty B_n$ is orthonormal?

Thanks

Answer 1

Together with the requirement of the comment, this cannot be achieved. Consider, e.g., $$A_1 = \{(1,0)\}, A_2 = \{(1,1)\}$$ in $\mathbb{R}^2$. Then, $A$ is linear independent and by $span(A_n) = span(B_n)$, you can only scale the vectors in $A_1$ and $A_2$. Thus, they will never become orthogonal.