DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-01

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The standard product norms on a finite product of normed spaces

Definition

Let $n \in N$ with $n \geq 1$ , and let $E_{0}, \dots, E_{n - 1}$ be normed spaces. On the Cartesian product $E : = \prod_{k < n} E_{k}$ write $x = (x_{0}, \dots, x_{n - 1})$ . The three standard product norms on $E$ are

$∥ x ∥_{\max} : = \max_{k < n} ∥ x_{k} ∥,$ $∥ x ∥_{1} : = \sum_{k < n} ∥ x_{k} ∥,$ $∥ x ∥_{2} : = (\sum_{k < n} ∥ x_{k} ∥^{2})^{1 / 2} .$

The finite sum is that of Finite sums and finite products, by recursion, and the symbol $n$ in the displayed upper bound is the canonical natural of The canonical natural $ι (n) = n \cdot 1_{F}$ of a field when it is read inside real inequalities.

Remarks

The hypothesis $n \geq 1$ is used only for the maximum norm, since the maximum of an empty family is not defined here.
For $n = 1$ all three product norms reduce to the original norm on the single factor.

Depends on

Used by

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Theo Buhler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)
Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)