Bound on $a_i$ given $v=a_1v_1+...+a_nv_n$

Question

Suppose $v_1,...,v_n$ is a basis of $V$ (an inner product space over $\textbf F$), and $f_1,...,f_n$ are functions $V\rightarrow \textbf F$ such that $$v=f_1(v)v_1+\cdots +f_n(v)v_n \ \ \ (v\in V)$$ Find $c\ge 0$ such that $$\lvert f_i(v)\rvert \le c\lvert\lvert v\rvert \rvert \ \ \ (i=1,...,n)$$ for all $v\in V$. All I have is that if $v_1,...,v_n$ is orthonormal we can take $c=1$