Lp norm (source code)
= Lp norm
{c}
{title2=$\lVert\mathord\cdot\rVert_p$}
An <Lp norm> measures the size of a <measurable function> by averaging a power of its <absolute value> with respect to a specified <measure>. For $1\leq p<\infty$,
$$
\lVert f\rVert_p=\left(\int |f|^p\,d\mu\right)^{1/p},
$$
when the <integral> is finite. The <exponent> $p$ controls how strongly large values influence the result. The <Lp space> identifies <functions> equal <almost everywhere>, so this becomes a <norm> on the corresponding <equivalence classes>. For $p=\infty$, use the <essential supremum> of $|f|$, as defined by the <L-infinity norm>. <Measure Theory by John K. Hunter/Lp spaces (Chapter 7 of Measure Theory)>[Hunter, Measure Theory, Definitions 7.1 and 7.4: Lp spaces and norms]{ref}.
For a two-coordinate <vector> $(3,4)$, treat the coordinates as a <function> on two inputs with <counting measure>. Then $\lVert(3,4)\rVert_1=3+4=7$, $\lVert(3,4)\rVert_2=\sqrt{9+16}=5$, and $\lVert(3,4)\rVert_\infty=4$. The middle case is the <Euclidean norm>; the first adds the coordinate magnitudes, and the last takes their largest value. These are different answers to different questions about size, all satisfying the <triangle inequality>.
If instead each input has <probability> $1/2$, the same values give $\lVert f\rVert_1=3.5$ and $\lVert f\rVert_2=\sqrt{12.5}$, while $\lVert f\rVert_\infty=4$ remains unchanged. Thus the <measure> is not an optional label. On a <probability space>, the finite <Lp norms> are nondecreasing in $p$ whenever they are finite, by <Hölder's inequality>; the unnormalized two-coordinate example above decreases instead. Statements comparing different <exponents> must specify their <measure> convention.
Why are these sizes useful? If $e$ is an error <function>, its <L1 norm> measures aggregate <absolute error>, its <L2 norm> measures the square root of aggregate <quadratic loss>, and its <L-infinity norm> bounds error <almost everywhere>. Concentrating a large error on a <set> of small positive <measure> can give a modest finite <Lp norm> but a large <L-infinity norm>. The three-point example and diagram in <L-infinity norm> show this explicitly, including why a <null set> behaves differently from a rare event.
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