LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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The standard finite product norms are equivalent

Statement

Let $n \geq 1$ and let $E = \prod_{k < n} E_{k}$ be a finite product of normed spaces, equipped with the three standard product norms of The standard product norms on a finite product of normed spaces. Then for every $x \in E$ , $∥ x ∥_{\max} \leq ∥ x ∥_{2} \leq ∥ x ∥_{1} \leq n ∥ x ∥_{\max} .$ Consequently these three norms are equivalent.

Facts & Assumptions

Given: A natural number $n \geq 1$ , normed spaces $E_{0}, \dots, E_{n - 1}$ , and a point $x = (x_{0}, \dots, x_{n - 1}) \in \prod_{k < n} E_{k}$ .

[L1]

The three product norms are $∥ x ∥_{\max} = \max_{k < n} ∥ x_{k} ∥$ , $∥ x ∥_{1} = \sum_{k < n} ∥ x_{k} ∥$ , and $∥ x ∥_{2} = (\sum_{k < n} ∥ x_{k} ∥^{2})^{1 / 2}$ (The standard product norms on a finite product of normed spaces).

[L2]

In $R^{n}$ , the Euclidean norm is bounded above by the $1$ -norm, and Cauchy-Schwarz holds (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

Proof

technique · direct

1.1

For each coordinate, $∥ x_{k} ∥^{2}$ is one of the nonnegative summands in $\sum_{j < n} ∥ x_{j} ∥^{2}$ , so $∥ x_{k} ∥ \leq ∥ x ∥_{2}$ for every $k$ and therefore $∥ x ∥_{\max} \leq ∥ x ∥_{2}$ .

L1algebra

1.2

Applying the scalar inequality $∥ (a_{k}) ∥_{2} \leq ∥ (a_{k}) ∥_{1}$ from [L2] to the nonnegative real tuple $(∥ x_{0} ∥, \dots, ∥ x_{n - 1} ∥)$ gives $∥ x ∥_{2} \leq ∥ x ∥_{1}$ .

L1L2

1.3

Since every coordinate norm is at most $∥ x ∥_{\max}$ , summing $n$ such bounds gives $∥ x ∥_{1} \leq n ∥ x ∥_{\max}$ .

L1algebra

2.1

The three displayed inequalities of steps 1.1, 1.2, and 1.3 are exactly the comparison constants needed for equivalence of the three norms.

step 1.1step 1.2step 1.3∎

Depends on

Used by

Finite products of Banach spaces are Banach Theorem

Dependency tree · two levels

26 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)