 Solution

In the space of tempered distributions, this Fourier transform convention gives $\hat{1}=2\pi \delta$ and $\hat{\delta }=1$. ThereforeThe constant distribution and the Dirac delta distribution are linearly independent: testing away from zero first determines the constant coefficient, and then testing at zero determines the delta coefficient. Hence $c=\lambda$ and $\lambda c=2\pi$, giving