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为什么无穷范数是取的最大值

2023-06-10 14:50 作者:~Sakuno酱  | 我要投稿

考虑这样的一个极限

%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%5Csqrt%5Bn%5D%7B%5Csum_%7Bi%7D%7Cx_i%7C%5En%7D


我们可以先考虑一个比较简单的情况 

%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%5Csqrt%5Bn%5D%7Ba%5En%2Bb%5En%7D   其中 a%3E0%2C%20b%3E0 假如 a%3Db 

原式就等于 %5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%5Csqrt%5Bn%5D%7B2a%5En%7D%3D%5Clim_%7Bn%20%5Cto%20%5Cinfty%7Da%5Csqrt%5Bn%5D%7B2%7D%20%3D%20a


如果 a%20%5Cne%20b 不失一般性,让 a%3Eb

%5Csqrt%5Bn%5D%7Ba%5En%2Bb%5En%7D%3Da%5Csqrt%5Bn%5D%7B1%2B(%5Cfrac%7Bb%7D%7Ba%7D)%5En%7D

%3Da%5Cexp(%5Cfrac%7B%5Cln%20(1%2B(%5Cfrac%7Bb%7D%7Ba%7D)%5En)%7D%7Bn%7D)%20

%3Da%5Cexp(%5Cfrac%7B%5Cln%20(1%2B(%5Cfrac%7Bb%7D%7Ba%7D)%5En)%7D%7B(%5Cfrac%7Bb%7D%7Ba%7D)%5En%7D%20%5Cfrac%7B(%5Cfrac%7Bb%7D%7Ba%7D)%5En%7D%7Bn%7D)%20

因为 %5Cfrac%7Bb%7D%7Ba%7D%20%3C%201 所以 %5Clim_%7Bn%20%5Cto%20%5Cinfty%7D(%5Cfrac%7Bb%7D%7Ba%7D)%5En%3D0 回顾下重要极限 %5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%5Cfrac%7B%5Cln%20(1%2Bx)%7D%7Bx%7D%3D1

所以

%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%20%5Csqrt%5Bn%5D%7Ba%5En%2Bb%5En%7D%3D%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20a%5Cexp(%5Cfrac%7B%5Cln%20(1%2B(%5Cfrac%7Bb%7D%7Ba%7D)%5En)%7D%7Bn%7D)%20%3Da


假如%5Cmathrm%7Bx%7D%3D(x_1%2Cx_2)是一个二维的向量

我们再定义 d_n(%5Cmathrm%7Bx%7D)%3D%20%5Csqrt%5Bn%5D%7B%7Cx_1%7C%5En%20%2B%7Cx_2%7C%5En%7D

再定义 d_%7B%5Cinfty%7D(%5Cmathrm%7Bx%7D)%3D%5Clim_%7Bn%20%5Cto%20%5Cinfty%7Dd_n(%5Cmathrm%7Bx%7D)

便有了 d_%7B%5Cinfty%7D(%5Cmathrm%7Bx%7D)%3D%5Cmax%5C%7Bx_1%2Cx_2%5C%7D


再考虑更高维的情况 高维的情况可以转化成二维的情况处理

假设有 p

这个时候我们就不能只考虑最大值了,因为可能有多个最大值

所以不失一般性,我们可以假设  %5Cmax_%7B1%5Cle%20i%20%5Cle%20p%7D%5C%7B%7Cx_i%7C%5C%7D%3D%7Cx_1%7C%3D%7Cx_2%7C%3D...%3D%7Cx_m%7C

%5Cmax_%7Bi%7D%5C%7B%7Cx_i%7C%5C%7D%3D0 时候是一个平凡情况

于是

%20%5Csqrt%5Bn%5D%7B%5Csum_%7Bi%7D%7Cx_i%7C%5En%7D%3D%20%5Csqrt%5Bn%5D%7Bm%7Cx_1%7C%5En%2B%5Csum_%7Bi%20%3Em%7D%7Cx_i%7C%5En%7D

%3D%7Cx_1%7C%5Csqrt%5Bn%5D%7Bm%2B%5Csum_%7Bi%3Em%7D%7C%5Cfrac%7Bx_i%7D%7Bx_1%7D%7C%5En%7D

这里直接用两边夹法则

%5Csqrt%5Bn%5D%7Bm%7D%5Cle%20%5Csqrt%5Bn%5D%7Bm%2B%5Csum_%7Bi%3Em%7D%7C%5Cfrac%7Bx_i%7D%7Bx_1%7D%7C%5En%7D%20%5Cle%20%5Csqrt%5Bn%5D%7Bm%2Bp-m%7D

可以计算出假如x_1取得最大值有

%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%5Csqrt%5Bn%5D%7B%5Csum_%7Bi%7D%7Cx_i%7C%5En%7D%3D%7Cx_1%7C



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