Suppose I have a Hilbert space $H$ with countable orthonormal basis $\{e_n \}_{n\in\mathbb{Z}_{\geq 0}}$. Every orthonormal basis is an example of an unconditional Schauder basis, meaning if I write $x\in H$ as $$ x=\sum_{n\geq 0} c_n e_n, \quad |c_n|<\infty $$ then I can rearrange the terms in the sum in any way I like and the series will still converge to $x$.
Now, suppose I split the coefficients $c_n$ into finite sums of another finite set of coefficients, that is for each $n$ I write $$ c_n=\sum_{m=0}^{K_n}b_{nm}, \quad |b_{nm}|<\infty,\quad K_n<\infty $$
Question: Can I still reorder the resulting series in any way I like? To give an example of what I mean, suppose $K_0=1$, $K_j=0$, $j>1$. So I have $$ x=(b_{00}+b_{01})e_0 +\sum_{n\geq 1}c_1 e_1 $$ Can I then move $b_{01} e_0$ to any position I like and write $$ x=b_{00} e_0 +c_1 e_1 + \dots + b_{01} e_0 + \dots\quad $$ or is it crucial that I keep all $e_n$ together?
Any vector in $H$ is uniquely defined by the coordinate projections $\langle e_n,x\rangle$. Since each $K_n$ is finite one can reorder the basis vectors in any desired way and this will not affect the coordinate values.