= Minkowski inequality for error aggregation
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{title2=$\lVert f-a\rVert_p\leq\lVert f-b\rVert_p+\lVert b-a\rVert_p$}
Suppose an approximation $a$ is compared with a target $f$ through an intermediate approximation $b$, with both error <functions> in the same <Lp space>. The identity $f-a=(f-b)+(b-a)$ and the <Minkowski inequality> combine two error budgets without any <independence> assumption. The <L1 norm> controls aggregate <absolute error>, the <L2 norm> controls the square root of aggregate <quadratic loss>, and the <L-infinity norm> controls the largest error <almost everywhere>. For equally weighted samples with the <probability measure> $1/n$ on each coordinate, the <L2 norm> is the <root mean square> error; the inequality still applies when this normalization is used consistently.
For example, two signed error <vectors> $e=(3,0)$ and $r=(0,4)$ have <Euclidean norms> $3$ and $4$, while $\lVert e+r\rVert_2=5\leq7$. Reinforcing errors $e=(3,0)$ and $r=(4,0)$ attain the bound $7$; opposing errors $e=(3,0)$ and $r=(-3,0)$ cancel completely. The <Minkowski inequality> always allows reinforcement. A smaller combined budget requires additional information about the errors, not just their separate <norms>.
Iteration gives $\lVert\sum_{k=1}^N e_k\rVert_p\leq\sum_{k=1}^N\lVert e_k\rVert_p$ for every finite sum. For an infinite <series>, if all $e_k$ belong to one fixed <Lp space> with $1\leq p\leq\infty$ and $\sum_{k=1}^\infty\lVert e_k\rVert_p<\infty$, its <partial sums> form a <Cauchy sequence> in the full <Lp norm>. <Completeness> of this <Banach space> then gives <norm convergence> to an element of the same <Lp space>. A finite sum bound alone does not establish convergence, and <completeness> is an additional ingredient beyond the <Minkowski inequality>. <Measure Theory by John K. Hunter/Lp spaces (Chapter 7 of Measure Theory)>[Hunter, Measure Theory, §7.4, Lemma 7.9, pp. 81–82, and Theorem 7.10, pp. 82–83: norm summability and completeness]{ref}
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