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  • 29595959

    证明:(1)因为$X_0$为$X$的真子空间,于是存在${x_1} in Xackslash {X_0}$,记$$d = mathop {inf }limits_{x in {X_0}} left| {x - {x_1}} ight|$$

    (2)因为$X_0$是闭的,故$d>0$,否则存在${x_n} in {X_0}$,且$left| {{x_n} - {x_1}} ight| o 0$,再由$X_0$是闭的推出${x_1} in {X_0}$矛盾

    (3)不妨设$varepsilon  < 1$,则有$frac{d}{{1 - varepsilon }} > d$,由下确界的定义知,存在${x_2} in {X_0}$,使得[left| {{x_2} - {x_1}} ight| < frac{d}{{1 - varepsilon }}]

    (4)令${x_0} = frac{{{x_1} - {x_2}}}{{left| {{x_1} - {x_2}} ight|}}$,则$left| {{x_0}} ight| = 1$,对于任何$x in {X_0}$,注意到${x_2} in {X_0}$,我们有

    egin{align*}
    left| {x - {x_0}} ight|& = left| {x - frac{{{x_1} - {x_2}}}{{left| {{x_1} - {x_2}} ight|}}} ight| = frac{1}{{left| {{x_1} - {x_2}} ight|}}left| {left( {left| {{x_1} - {x_2}} ight|x + {x_2}} ight) - {x_1}} ight|\&
    ge frac{1}{{left| {{x_1} - {x_2}} ight|}} cdot d > 1 - varepsilon
    end{align*}

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  • 原文地址:https://www.cnblogs.com/ly758241/p/3810297.html
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