Orthonormal sequence

An orthonormal sequence $(e_n)$ satisfies $⟨e_n,e_m⟩=0$ for $n\ne m$ and $\Vert e_n\Vert =1$.

A Hilbertian basis is a complete orthonormal family: its closed linear span is the whole Hilbert space, equivalently every vector is the norm-convergent sum of its Fourier coefficients against the basis.

For a Hilbertian basis $(e_i)$,

For an orthonormal sequence $(e_n)$ in a Hilbert space,