of Lp, 0 < p ≤ ∞ For some authors, notably Royden, the Riesz-Fischer Theorem is the result that is complete: that every Cauchy sequence of functions in converges to a function in under the metric induced by the p-norm. The proof below is based on the convergence theorems for the Lebesgue integral; the result can also be obtained for by showing that every Cauchy sequence has a rapidly converging Cauchy sub-sequence, that every Cauchy sequence with a convergent sub-sequence converges, and that every rapidly Cauchy sequence in converges in When the Minkowski inequality implies that the Lp space is a normed space. In order to prove that is complete, i.e. that is a Banach space, it is enough (see e.g. Banach space#Definition) to prove that every series of functions in such that converges in the -norm to some function For the Minkowski inequality and the monotone convergence theorem imply that is defined –almost everywhere and The dominated convergence theorem is then used to prove that the partial sums of the series converge to f in the -norm, The case requires some modifications, because the p-norm is no longer subadditive. One starts with the stronger assumption that and uses repeatedly that The case reduces to a simple question about uniform convergence outside a -negligible set. See also References . . . . Fourier series Theorems in real analysis