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    证明:令$d = mathop {inf}limits_{y in M} left| {x - y} ight|$,由下确界的定义知,存在${x_n} in M$,使得[mathop {lim }limits_{n o infty } left| {{x_n} - x} ight| = d]

       下面我们证明$left{ {{x_n}} ight}$是基本列.由平行四边形公式知[{left| {{x_m} - x} ight|^2} + {left| {{x_n} - x} ight|^2} = 2left( {{{left| {frac{{{x_m} + {x_n}}}{2} - x} ight|}^2} + {{left| {frac{{{x_m} - {x_n}}}{2}} ight|}^2}} ight)]由于$frac{{{x_m} + {x_n}}}{2} in M$,则$left| {frac{{{x_m} + {x_n}}}{2} - x} ight| ge d$,从而可知[0 le frac{1}{2}{left| {{x_m} - {x_n}} ight|^2} le {left| {{x_m} - x} ight|^2} + {left| {{x_n} - x} ight|^2} - 2{d^2}]令$n,m o infty $,则$left| {{x_m} - {x_n}} ight| o 0$,所以$left{ {{x_n}} ight}$是基本列

    又由于$M$为$f{Hilbert}$空间$X$的闭子空间,则存在${x_0} in M$,使得${x_n} o {x_0}$,此时[left| {x - {x_0}} ight| = mathop {lim }limits_{n o infty } left| {x - {x_n}} ight| = d]

       下面我们证明$x - {x_0} ot M$.由线性子空间的定义知,对任意的$z in M,z e 0$,以及$lambda  in K$,有${x_0} + lambda z in M$,于是[left| {x - left( {{x_0} + lambda z} ight)} ight| ge d]所以有[{left| {left( {x - {x_0}} ight) - lambda z} ight|^2} = {left| {x - {x_0}} ight|^2} - 2{mathop{ m Re} olimits} left( {overline lambda  left( {x - {x_0}} ight),z} ight) + {left| lambda   ight|^2}{left| z ight|^2} ge {d^2}]令$lambda  = frac{{left( {x - {x_0},z} ight)}}{{{{left| z ight|}^2}}}$,则有

    egin{align*}
    {left| {x - {x_0}} ight|^2} - 2frac{{{{left| {left( {x - {x_0},z} ight)} ight|}^2}}}{{{{left| z ight|}^2}}} + frac{{{{left| {left( {x - {x_0},z} ight)} ight|}^2}}}{{{{left| z ight|}^2}}}\
    = {left| {x - {x_0}} ight|^2} - frac{{{{left| {left( {x - {x_0},z} ight)} ight|}^2}}}{{{{left| z ight|}^2}}} ge {d^2}
    end{align*}

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  • 原文地址:https://www.cnblogs.com/ly758241/p/3806787.html
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