The $p$-norms $\lVert x\rVert_p$ for rational $p \ge 1$, and $\lVert x\rVert_\infty$

Let $n \in \mathbb{N}$ and let $\mathbb{R}^{n}$ be the function space of The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$, with $x_k := x(k)$ for $k < n$.

The $p$-norm, for a rational exponent $p \ge 1$

Let $p \in \mathbb{Q}$ with $p \ge 1$. For $x \in \mathbb{R}^{n}$ put

$\lVert x\rVert_p \;:=\; \Bigl(\sum_{k<n} |x_k|^{p}\Bigr)^{1/p},$

where $|\cdot|$ is the absolute value (Absolute value in an ordered field), the sum is the finite sum of Finite sums and finite products, by recursion, and both powers are the rational powers of Rational powers $a^r$ of a positive base.

Every power written here is defined. Each base $|x_k|$ is a nonnegative real and $p > 0$, so $|x_k|^{p}$ is given by Rational powers $a^r$ of a positive base for $|x_k| > 0$ and by its supplementary clause $0^{p} = 0$ for $|x_k| = 0$; the sum of these nonnegative terms is nonnegative (Laws of finite sums and finite products clause 4), and $1/p$ is a positive rational, so the outer power is defined for the same two reasons. The value does not depend on which representative of $p$ or of $1/p$ is used (Rational powers do not depend on the representative).

The exponent is a rational, and that is not a matter of taste. Rational powers $a^r$ of a positive base supplies $a^{r}$ for a nonnegative base and a rational exponent only; real exponents do not exist at this point in the reading order, and Why real exponents are deferred on the rational-powers page records exactly why. This is also why the published Minkowski inequality Minkowski's inequality for finite sums (rational exponent), which is what makes the triangle inequality work below, is itself stated for rational $p \ge 1$. No statement on this page is written for $p$ ranging over a real interval.

The maximum norm

For $n \ge 1$ and $x \in \mathbb{R}^{n}$ put

$\lVert x\rVert_\infty \;:=\; \max\{\, |x_k| \;:\; k < n \,\},$

the maximum of a nonempty finite set of reals, which exists and is one of its elements (Every nonempty finite set of reals has a maximum and a minimum, Maximum and minimum of a set).

The hypothesis $n \ge 1$ is required and propagates. At $n = 0$ the set $\{|x_k| : k<n\}$ is empty and has no maximum (Maximum and minimum of a set). This is the same restriction the published $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it carries, for the same reason, and every statement on this page that mentions $\lVert\cdot\rVert_\infty$ inherits it. The $p$-norms for rational $p \ge 1$ carry no such restriction: at $n = 0$ each is the empty sum raised to a positive rational power, hence $0$.

The three cases the rest of the page uses

• $\lVert x\rVert_1 = \sum_{k<n}|x_k|$, since $t^{1} = t$ for $t \ge 0$ (Laws of rational exponents, and $0^{1} = 0$ by the supplementary clause).
• $\lVert x\rVert_2 = \bigl(\sum_{k<n}|x_k|^{2}\bigr)^{1/2} = \sqrt{\sum_{k<n}x_k^{2}}$, which is exactly the Euclidean norm of The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$: the exponent $2$ agrees with the integer power, so $|x_k|^{2} = x_k^{2}$ (Basic properties of the absolute value), and $t^{1/2}$ is the unique nonnegative square root of $t$, which is $\sqrt{t}$ (Rational powers $a^r$ of a positive base, Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$; the positives are $\{x^2 : x \neq 0\}$). The two notations denote the same function and no second Euclidean norm is introduced.
• $\lVert x\rVert_\infty$ as above, for $n \ge 1$.

That each of these is a norm in the sense of A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, and that the metrics they induce are exactly the published $d_1$, $d_2$ and $d_\infty$ of $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it, is Each $\lVert\cdot\rVert_p$ is a norm on $\mathbb{R}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_\infty$ of the published metric-spaces page; it is proved there and is not assumed here.

• Why $p \ge 1$. The triangle inequality for $\lVert\cdot\rVert_p$ is Minkowski's inequality, and Minkowski's inequality for finite sums (rational exponent) is stated for rational $p \ge 1$. For $0 < p < 1$ the displayed expression is still defined but is not a norm on $\mathbb{R}^{n}$ for $n \ge 2$; nothing on this page asserts anything about that range, and the expression is never written with such an exponent.

• Monotonicity in the base is what makes the comparisons below work. For a fixed positive rational $r$ the map $a \mapsto a^{r}$ is strictly increasing on the positive reals (Monotonicity of $r \mapsto a^{r}$ and of $a \mapsto a^{r}$ clause 2), so an inequality between nonnegative sums passes through the outer power. That is the only property of rational powers used in the comparison chain of The finite and reverse triangle inequalities for a norm; and for $n \ge 1$ every norm $N$ on $\mathbb{R}^n$ satisfies $N(x) \le C\lVert x\rVert_1$ and is Lipschitz, hence continuous, for $d_2$.

• The subscript $\infty$ is a name, not a number. No arithmetic is performed with it, and $\lVert\cdot\rVert_\infty$ is not $\lVert\cdot\rVert_p$ for any exponent; it is a separately defined function that happens to sit at the end of the family. This is the same refusal to extend $\mathbb{R}$ silently that Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length records for the interval notation.