\[\left\langle\mathscr{F}\left(\delta\right),\phi\right\rangle=\left\langle%
\delta,\mathscr{F}(\phi)\right\rangle=\left\langle\delta,\frac{1}{\sqrt{2\pi}}%
\int^{\infty}_{-\infty}\phi(t){\mathrm{e}}^{\mathrm{i}xt}\,\mathrm{d}t\right%
\rangle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\phi(t)\,\mathrm{d}t=\frac%
{1}{\sqrt{2\pi}}\left\langle 1,\phi\right\rangle,\]