$(12苏大四)$设$f\left( x \right) \in {C^1}\left( { - \infty , + \infty } \right)$，且

∫+∞?∞[f(x)2+f′2(x)]dx=1

$(2)$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

∫+∞0(|u(x)|2+∣∣u′(x)∣∣2)dx<+∞

$(1)$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

Supx∈[0,+∞)|u(x)|≤C(∫+∞0|u(x)|2+∣∣u′(x)∣∣2dx)12