Let be $L_\infty(\mathbb{R})$, $L_1(\mathbb{R})$, prove that the dual space of $L_\infty(\mathbb{R})$, $[L_\infty(\mathbb{R})]^*$, can't be identified with $L_1(\mathbb{R})$, sp in other words: $$\forall L\in [L_\infty(\mathbb{R})]^* \ \nexists g \in L_1(\mathbb{R}) : Lf=\int_\mathbb{R}fgdx, \ f\in L_\infty(\mathbb{R})$$
Can you please help with this one? I don't know how to prove it.
A note: I define the spaces $L_{\infty}(\mathbb{R})$, $L_1(\mathbb{R})$, the norms $||f||_\infty$, $||f||_1$ of a lebesgue measurable functions and the lebesgue measure on $L$ lebesgue $\sigma$-algebra as such: $$|*|:L\rightarrow[0,+\infty]$$
$$||f||_\infty=\inf\{t\geq0 : |\{|f|>0\}|=0\}$$ $$||f||_1=\int_\mathbb{R}|f|dx$$
$$L_{\infty}(\mathbb{R})=\{f:X\rightarrow\mathbb{K} \ | \ ||f||_\infty<+\infty\}$$ $$L_{1}(\mathbb{R})=\{f:X\rightarrow\mathbb{K} \ | \ ||f||_1<+\infty\}$$
where every element of $L_{\infty}(\mathbb{R})$ or $L_{1}(\mathbb{R})$ is a equivalence class [f], denoted as $f$ to simplify, defined as: $$[f]=\{g:X\rightarrow\mathbb{K} \ | f=g \ lebesgue-a.e.\}$$