Euclidean Convergence (ユークリッド収束) (Definition III.6.5)
- Euclidean Convergence (ユークリッド収束) (Definition III.6.5) #Card
- On $\mathbb{R}^m$, the metrics $d(x,y) = \sqrt{\sum_k |x_k - y_k|^2}$ (Euclidean), $s(x,y) = \sum_k |x_k - y_k|$, $t(x,y) = \max_k |x_k - y_k|$ satisfy $t \leq d \leq s \leq m \cdot t$, hence are equivalent: $t_t = t_d = t_s$.
$\mathbb{R}^m = {(x_1, \dots, x_m) : x_k \in \mathbb{R}}$ is a Euclidean space of dimension $m$. The functions $$d(x,y) := \sqrt{\textstyle\sum_{k=1}^m |x_k - y_k|^2}, \quad s(x,y) := \textstyle\sum_{k=1}^m |x_k - y_k|, \quad t(x,y) := \max_{1 \le k \le m} |x_k - y_k|$$ are metrics; $d$ is the Euclidean metric.
- (III.6.8): $t(x,y) \leq d(x,y) \leq s(x,y) \leq m \cdot t(x,y)$、よって 3 つの距離は同値で同じ収束を定める: $t_t = t_d = t_s$。
- 収束は座標ごと: $x \in \lim \mathcal{F} \iff x_k \in \lim_\nu p_k[\mathcal{F}]$ for each $1 \leq k \leq m$（$p_k$ は第 $k$ 射影）。
EuclideanSpace ℝ (Fin m)；同値性は Pi 上の dist 比較（dist_pi_le_iff など）