= Riesz-Fischer theorem for an orthonormal system
{c}
{title2=$\sum|\alpha_n|^2<\infty\Longrightarrow\sum\alpha_ne_n\text{ converges}$}
For an orthonormal family in a <Hilbert space>, the squared <norm> of every finite partial sum is the sum of squared coefficients. Square summability therefore makes the net of finite partial sums Cauchy, proving unconditional <convergence> by <completeness>. If the orthonormal family is complete, <Bessel's inequality> makes the coefficients of each vector square summable, and the residual after summing is orthogonal to the whole family, hence zero. This recovers the orthonormal expansion and <Parseval identity>.
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