how to show that the operator $T(t)$ defined on $\mathbb{X}= \left\{ u \in \mathbf{C}\left(\mathbb{R}\right): \lim\limits_{ x \rightarrow \pm \infty} u(x) = 0\right\}$ endowed with the norm $$\|u\|_{\infty} = \sup_{-\infty <x<+\infty} |u(x)|$$ by :
$T(t)f(x) = \frac{1}{\sqrt{4\pi t}} \int_{-\infty}^{+\infty} e^{-\frac{|x-y|^{2}}{4t}}f(y) \, dy$. $(t >0)$
is not compact ?
I can't build a suitable sequence $(\varphi_{n})_{n}$ such that $T(\varphi_{n})(x)$ diverges on $\mathbb {X}$