2.23

Assume to the contrary that there does exist an $I \in L^1(\mathbb{R}^d)$ such that: $$(f*I) = f \hspace{0.25cm} \forall f \in L^1(\mathbb{R}^d)$$ It follows from the latter parts of exercise 21 that: $$\hat{f}(\xi) = \widehat{(f*I)}(\xi) = \hat{f}(\xi)\hat{I}(\xi)$$ Thus, since $\hat{f}(\xi)$ need not be zero, we have that $\hat{I}(\xi) = 1 \hspace{0.25cm} \forall \xi$. I.e. $\lim_{\xi \to \infty} \hat{I}(\xi) = 1$. This contradicts the Riemann-Lebesgue Lemma. Therefore, $I \notin L^1(\mathbb{R}^d)$.