Convolution of $\delta$ and $f$, in $L^1$ space

Question

How to show that it doesn't exist $\delta \in L^1(\mathbb{R})$ such as $\delta *f=f$ in $L^1(\mathbb{R})$ for all $f \in L^1(\mathbb{R})$.

$\delta * f$ is the convolution of $\delta$ and $f$.

Someone could help me to prove this ?

Answer 1

Use $ \widehat{\delta} {\cdot} \widehat{f} =\widehat{\delta \ast f} = \widehat{f}$ and the Riemann-Lebesgue lemma to get a contraction.