Is there a theorem of this form?
Given a vector $x$ and an ONB $\{v_1, v_2, \ldots, v_n\}$, $\exists \, c>0$ such that $\langle x, v_j \rangle \ge c ||x||$ for some $j$. In other words, when you take the orthogonal decomposition of a vector, there has to be at least one coefficient somewhat large.
I think some such result has to exist, but I'm hoping that someone might be able to tell me more about the constant $c$ and how it is related to the ONB.
If $x=0$ everything is clear. Take any $x\in H\setminus \{0\}$, then $$ x=\sum_{i=1}^n\langle x, v_i\rangle v_i $$ Now we argue towards a contradiction. Assume for all $c>0$ and $i\in\mathbb{N}_n$ we have $\langle x,v_i<c\Vert x\Vert$. Then $$ \Vert x\Vert \leq \sum_{i=1}^n|\langle x, v_i\rangle| \Vert v_i\Vert \leq \sum_{i=1}^n c\Vert x\Vert =nc\Vert x\Vert $$ Set $c=(2n)^{-1}$ and recall that $x\neq 0$, to get a contradiction.