介紹
![閔科夫斯基不等式](/img/8/4ca/wZwpmL0YjMyQTM4QDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL0gzL4czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![閔科夫斯基不等式](/img/6/46c/wZwpmLzMTO1cTNwIDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLyAzL1YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
在數學中, 閔可夫斯基不等式(Minkowskiinequality)表明L空間是一個賦范向量空間。設S是一個度量空間, ,那么 ,我們有:
![閔科夫斯基不等式](/img/d/0d0/wZwpmLwgjM0gDM5MzM3UzM1UTM1QDN5MjM5ADMwAjMwUzLzMzL4IzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![閔科夫斯基不等式](/img/a/970/wZwpmLyYzM2MjMxYDM3UzM1UTM1QDN5MjM5ADMwAjMwUzL2AzLzAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![閔科夫斯基不等式](/img/2/088/wZwpmLyYzN0ATOzMjN2UzM1UTM1QDN5MjM5ADMwAjMwUzLzYzL3EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
如果 ,等號成立若且唯若,或者g=kf。
![閔科夫斯基不等式](/img/6/7bf/wZwpmL4YTO0cjN4gTN2UzM1UTM1QDN5MjM5ADMwAjMwUzL4UzLwYzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
閔可夫斯基不等式是 中的三角不等式。它可以用赫爾德不等式來證明。和赫爾德不等式一樣,閔可夫斯基不等式取可數測度可以寫成序列或向量的特殊形式:
![閔科夫斯基不等式](/img/9/9c1/wZwpmLwUTM1kTOwIDN3UzM1UTM1QDN5MjM5ADMwAjMwUzLyQzL3AzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![閔科夫斯基不等式](/img/a/30a/wZwpmL4QjN0QjN3QzM3UzM1UTM1QDN5MjM5ADMwAjMwUzL0MzLwQzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![閔科夫斯基不等式](/img/5/847/wZwpmL4ITOzMzMwQjN2UzM1UTM1QDN5MjM5ADMwAjMwUzL0YzL2gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![閔科夫斯基不等式](/img/4/0e2/wZwpmL0cDMwMjMxMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL0YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![閔科夫斯基不等式](/img/3/f4b/wZwpmL1UDO0ATO1ADO3EDN0UTMyITNykTO0EDMwAjMwUzLwgzLxMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
將所有實數(n為S的維數)改成複數同樣成立。值得指出的是,如果 ,則 可以變為 。
積分形式的證明
![閔科夫斯基不等式](/img/5/5fc/wZwpmL4ATMxgzNxEjN2UzM1UTM1QDN5MjM5ADMwAjMwUzLxYzL0UzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
我們考慮 的p次冪:
![閔科夫斯基不等式](/img/4/ffe/wZwpmLzgDO4ITOzMzM3UzM1UTM1QDN5MjM5ADMwAjMwUzLzMzLyYzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
(用三角形不等式展開|f(x)+g(x)|)
![閔科夫斯基不等式](/img/c/6e6/wZwpmLyQTOxkDOzQDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL0gzL1MzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
(用赫爾德不等式)
![閔科夫斯基不等式](/img/1/fcf/wZwpmLyEDN0IDM5QzM3UzM1UTM1QDN5MjM5ADMwAjMwUzL0MzL4QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![閔科夫斯基不等式](/img/2/6cf/wZwpmLwgzNxQzNwAzN2UzM1UTM1QDN5MjM5ADMwAjMwUzLwczLzczLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![閔科夫斯基不等式](/img/1/e10/wZwpmLxAjN2UTN2QzN2UzM1UTM1QDN5MjM5ADMwAjMwUzL0czL4czLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
(利用 p=qp-q,因為)
![閔科夫斯基不等式](/img/5/be4/wZwpmL3QDOwcTMzQDM3UzM1UTM1QDN5MjM5ADMwAjMwUzL0AzLyMzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
現在我們考慮這個不等式序列的首尾兩項。首項除以尾項的最後一個因子,即得
![閔科夫斯基不等式](/img/2/ef2/wZwpmLwQDMzgTO3MDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLzAzL0MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
這正是我們所要的結論。對於序列的情形,證明是完全類似的。
參閱
•馬勒不等式