Let $V$ be a separable infinite dimensional Hilbert space over $\mathbb{C}$
Let $\{A_{i,j}\}_{i,j \in \mathbb{N}} \subset V$ be a matrix of linearly independent vectors of $V$
My question is if is it possible to orthonormalise matrix $A$, getting the matrix $B$ such that $$ \langle B_{i_1,j_1} , B_{i_2,j_2} \rangle = 0 $$ and such that $$ \forall i \in \mathbb{N}: \operatorname{Span}(\operatorname{Row}_i(A))=\operatorname{Span}(\operatorname{Row}_i(B)) $$
I have to orthonormalise a matrix and not a sequence so the question is very different from Gram-Schmidt in Hilbert space? mainly for the second condition. I cannot see no one anything in common. thanks.