在數學裏,積度量(product metric)是在兩個以上度量空間之笛卡爾積內的度量。n 個度量空間之笛卡爾積的積度量,可視為是將 n 個子空間的範數作為 n 維向量之各分量,取其 p-範數所得之值。 d p ( x 1 , … , x n ) = ‖ ( d 1 ( x 1 ) , … , d n ( x n ) ) ‖ p {\displaystyle d_{p}(\mathbf {x} _{1},\dots ,\mathbf {x} _{n})=\|(d_{1}(\mathbf {x} _{1}),\dots ,d_{n}(\mathbf {x} _{n}))\|_{p}} 令 ( X , d X ) {\displaystyle (X,d_{X})} 與 ( Y , d Y ) {\displaystyle (Y,d_{Y})} 為度量空間,且令 1 ≤ p ≤ + ∞ {\displaystyle 1\leq p\leq +\infty } 。 X × Y {\displaystyle X\times Y} 上之 p-積度量 d p {\displaystyle d_{p}} 定義為 對於 x 1 , x 2 ∈ X {\displaystyle x_{1},x_{2}\in X} 及 y 1 , y 2 ∈ Y {\displaystyle y_{1},y_{2}\in Y} , d p ( ( x 1 , y 1 ) , ( x 2 , y 2 ) ) := ( d X ( x 1 , x 2 ) p + d Y ( y 1 , y 2 ) p ) 1 / p {\displaystyle d_{p}\left((x_{1},y_{1}),(x_{2},y_{2})\right):=\left(d_{X}(x_{1},x_{2})^{p}+d_{Y}(y_{1},y_{2})^{p}\right)^{1/p}} for 1 ≤ p < ∞ ; {\displaystyle 1\leq p<\infty ;} d ∞ ( ( x 1 , y 1 ) , ( x 2 , y 2 ) ) := max { d X ( x 1 , x 2 ) , d Y ( y 1 , y 2 ) } . {\displaystyle d_{\infty }\left((x_{1},y_{1}),(x_{2},y_{2})\right):=\max \left\{d_{X}(x_{1},x_{2}),d_{Y}(y_{1},y_{2})\right\}.} 在歐氏空間裏,使用 L2 範數會在積空間裏產生歐幾里得度量;不過,選擇 p 的其他值也會形成其他拓撲等價的度量空間。在度量空間範疇(具有利普希茨常數為 1 的利普希茨映射)裏,使用上確界範數。 Deza, Michel Marie; Deza, Elena, Encyclopedia of Distances, Springer-Verlag: 83, 2009. Wikiwand in your browser!Seamless Wikipedia browsing. On steroids.Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.Wikiwand for ChromeWikiwand for Firefox
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.