decomposition of a vector in a Hilbert space

Question

this might be a basic question : If we consider a Hilbert space $H$ with the scalar product $\cdot$ and the norm induced by it : $\mid . \mid$, then, is it true that every vector $v$ in $H$ can be written as : $$v=\sum_{i=1}^{N}(v_j\cdot v)v_j$$ where $(v_j)_{j=1,...,N}$ is a orthonormal system in $H$.

Thanks,

Answer 1

If $\{v_1,v_2\}$ is orthonormal then so is $\{v_1\}$ and we cannot write $v_2$ as $\langle v_2 , v_1 \rangle v_1$ since $\langle v_2 , v_1 \rangle =0$. [Take $N=1$ and $\{v_1\}$ as your orthonormal set].