I have to prove the following statement:
$f\mapsto \phi(f)\vert\Omega\rangle$ is continuous, where $\phi$ is a scalar Wightman quantum field, $\Omega$ the vacuum state of the theory and $f$ a Schwartz-class function.
First of all, continuity means that $\Vert\phi(f_{n})\vert\Omega\rangle\Vert\to 0$ for $n\to\infty$, if $\Vert f_{n}\Vert\to 0$ for $n\to\infty$.
Per definition of the Wightman quantum field as a operator-valued distribution, I know that the map $f\mapsto \langle\Omega\vert\phi(f)\vert\Omega\rangle$ is a tempered distribution. Therefore I can write
$$\Vert\phi(f_{n})\Omega\Vert = \langle\Omega\vert\phi(f_{n})\phi(f_{n})\vert\Omega\rangle=\int\mathrm{d}^{4}x\mathrm{d}^{4}y\langle\Omega\vert\phi(x)\phi(y)\vert\Omega \rangle f_{n}(x)f_{n}(y)$$
My next step is to use the Källen-Lehmann representation: $$\Vert\phi(f_{n})\Omega\Vert=\int\mathrm{d}^{4}x\mathrm{d}^{4}y f_{n}(x)f_{n}(y)\int\mathrm{d}\rho(m^{2})\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}2\omega_{p}}e^{ip(x-y)}$$
Is this right up till know? Now I don`t know how to continue...According to a hint on the exercise, I am allowed to use, that $\rho(m^{2})$ is polynomially bounded measure and the Fourier transformation is linear and continuous....
