Riesz-Fischer theorem

For $1\le p\le \infty$, the Lebesgue space $L^p$ is complete in its usual norm.

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.