265656464

$(12苏大四)$设$f\left( x \right) \in {C^1}\left( { - \infty , + \infty } \right)$，且\[\int_{ - \infty }^{ + \infty } {\left[ {f{{\left( x \right)}^2} + {{f'}^2}\left( x \right)} \right]dx} = 1\]
证明：$(1)$$\lim \limits_{x \to \begin{array}{*{20}{c}}
\infty \end{array}} f\left( x \right) = 0$

$(2)$对任意$x \in \left( { - \infty , + \infty } \right)$，有$\left| {f\left( x \right)} \right| < \frac{{\sqrt 2 }}{2}$

$(08华师七)$设$u\left( x \right)$在$\left[ {0, + \infty } \right)$上连续可微，且
\[\int_0^{ + \infty } {\left( {{{\left| {u\left( x \right)} \right|}^2} + {{\left| {u'\left( x \right)} \right|}^2}} \right)dx} < + \infty \]证明：

$(1)$存在$\left[ {0, + \infty } \right)$上子列$\left\{ {{x_n}} \right\}$，使得${x_n} \to \infty $，且$u\left( {{x_n}} \right) \to 0\left( {n \to \infty } \right)$

$(2)$存在常数$C>0$，使得\[\mathop {Sup}\limits_{x \in \left[ {0, + \infty } \right)} \left| {u\left( x \right)} \right| \le C{\left( {\int_0^{ + \infty } {{{\left| {u\left( x \right)} \right|}^2} + {{\left| {u'\left( x \right)} \right|}^2}dx} } \right)^{\frac{1}{2}}}\]