Characterization of N-Cube
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Theorem
Let $\struct {\R^n, d}$ be a Euclidean $n$-Space equipped with the usual metric $d$.
Let $x, y \in \R^n$, where:
- $x = \tuple {x_1, x_2, \ldots, x_n}$
- $y = \tuple {y_1, y_2, \ldots, y_n}$
Let $R > 0$ be fixed.
Let:
- $\ds Q = \set {x, y \in \R^n: \sup_{x, y} \max_i \size {y_i - x_i} \le R}$
Then $Q$ is an $n$-cube.
Proof
For ease of presentation, denote:
- $y - x = r \in \R^n$
and:
- $y_j - x_j = r_j$
for $j = 1, 2, \ldots, n$.
\(\ds Q\) | \(=\) | \(\ds \set {r: \map {\sup_r} {\max_i \size {r_i} \le R} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {r: \map {\sup_r} {\map \max {\size {r_1}, \size {r_2}, \ldots, \size {r_n} } \le R} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {r: \map {\sup_\ell} {r_1 \in \closedint {-\ell} \ell, r_2 \in \closedint {-\ell} \ell, \ldots, r_n \in \closedint {-\ell} \ell, \ell \le R} }\) | where $\ell = \map \max {\size {r_1}, \size {r_2}, \ldots, \size {r_n} }$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \prod_{i \mathop = 1}^n \closedint {-R} R_i\) |
$\blacksquare$
Sources
- 1984: Gerald B. Folland: Real Analysis: Modern Techniques and their Applications : $\S \text P.6$