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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Explicit comparison constants for the standard norms on K^n

Example

Let $K \in {R, C}$ and let $x = (x_{0}, \dots, x_{n - 1}) \in K^{n}$ with $n \geq 1$ . Define

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

Then

$∥ x ∥_{\infty} \leq ∥ x ∥_{2} \leq ∥ x ∥_{1} \leq \sqrt{n} ∥ x ∥_{2} \leq n ∥ x ∥_{\infty} .$

In particular, the abstract norm-equivalence theorems on the A page can be read with explicit constants on these three standard coordinate norms.

Facts & Assumptions

Given: A field $K \in {R, C}$ , an integer $n \geq 1$ , and a vector $x \in K^{n}$ .

[L1]

On $R^{n}$ , the displayed $ℓ^{1}$ , Euclidean, and max formulas are the standard norms of The $p$ -norms $∥ x ∥_{p}$ for rational $p \geq 1$ , and $∥ x ∥_{\infty}$ , and every norm on $R^{n}$ is equivalent to every other (For $n \geq 1$ all norms on $R^{n}$ are equivalent).

[L2]

On finite-dimensional complex spaces every two norms are equivalent (All norms on a finite-dimensional complex normed space are equivalent).

Verification

technique · direct

1.1L1algebra

Since every summand $∣ x_{j} ∣^{2}$ is nonnegative and one of them equals $∥ x ∥_{\infty}^{2}$ , one has $∥ x ∥_{\infty}^{2} \leq \sum_{j < n} ∣ x_{j} ∣^{2} = ∥ x ∥_{2}^{2}$ , hence $∥ x ∥_{\infty} \leq ∥ x ∥_{2}$ . Also $∥ x ∥_{2}^{2} = \sum_{j < n} ∣ x_{j} ∣^{2} \leq {(\sum_{j < n} ∣ x_{j} ∣)}^{2} = ∥ x ∥_{1}^{2},$ so $∥ x ∥_{2} \leq ∥ x ∥_{1}$ .

2.1step 1.1algebra

By Cauchy-Schwarz for the vectors $(∣ x_{j} ∣)_{j < n}$ and $(1, \dots, 1)$ , $∥ x ∥_{1} = \sum_{j < n} ∣ x_{j} ∣ \leq {(\sum_{j < n} ∣ x_{j} ∣^{2})}^{1 / 2} {(\sum_{j < n} 1^{2})}^{1 / 2} = \sqrt{n} ∥ x ∥_{2} .$ Also $∣ x_{j} ∣ \leq ∥ x ∥_{\infty}$ for every $j$ , so $∥ x ∥_{1} \leq n ∥ x ∥_{\infty}$ .

3.1L1L2step 1.1step 2.1∎

Combining steps 1.1 and 2.1 yields the displayed chain. Thus [L1] and [L2] become concrete on these coordinate norms.

Remarks

The constants are sharp in the standard basis: $x = (1, \dots, 1)$ makes $∥ x ∥_{1} = n ∥ x ∥_{\infty}$ and $∥ x ∥_{1} = \sqrt{n} ∥ x ∥_{2}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

36 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

Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis (standard reference, not scraped)
Tomasz Kochanek, Functional analysis, Lecture 1 (standard reference, not scraped)