We assume that $g \in C_{b}(\mathbb{R})$. We consider the vector subspace
$$N_{g} = \left\{f \in L^{1}(\mathbb{R}) \mid f * g = 0\right\}$$
and the vector space $V_{g} = \operatorname{Vect}\left(T_{\alpha}(g)\right)_{\alpha \in \mathbb{R}}$ where $T_{\alpha}(g)(x) = g(x-\alpha)$.\\
Show that the codimension of $N_{g}$ in $L^{1}(\mathbb{R})$ is equal to the dimension of $V_{g}$.