Let an equivalence relation in $L^{\infty}=L^{\infty}(\mathbb{R}^n)$ as follows: we say that $\varphi \sim \varphi_1$ if $\varphi -\varphi_1= c$ a.e.
We identifying those $L^{\infty}$ functions that differ by a constant, and denote the equivalence class corresponding to $\varphi \in L^{\infty}$ by $\phi$. Now let $(\mathcal{L}^{\infty}(\mathbb{R}^n), \|\cdot\|_{\mathcal{L}^{\infty}})$ denote the $L^{\infty}$ modulo constants space endowed with the quotient norm, that is, $$\| \phi \|_{\mathcal{L}^{\infty}} = \inf \{ \|\varphi\|_{L^{\infty}}:\varphi \in \phi \}, \phi \in \mathcal{L}^{\infty}(\mathbb{R}^n).$$ Show that
(a) $(\mathcal{L}^{\infty}, \|\cdot\|_{\mathcal{L}^{\infty}})$ is Banach space ;
(b)$(\mathcal{L}^{\infty}, \|\cdot\|_{\mathcal{L}^{\infty}})$ is the dual space of $L^1_0$, where $L^1_0=L^1_0(\mathbb{R}^n)$ the subspace of $L^1(\mathbb{R}^n)$ functions with vanishing integral.
I think $ \mathcal{L}^{\infty} = L^{\infty }/_{\sim}$. For (a), I can't prove that $(\mathcal{L}^{\infty}, \|\cdot\|_{\mathcal{L}^{\infty}})$ is complete. For (b), I was trying to prove it by using Riesz Theorem similarly into proof $L^{\infty} $ is dual space of $L^1$ but there is no progress. Could you please prove or give me some references for the proof of this result?