Can the "evaluate" linear function be represented? Let $CB(\mathbb{R})$ be the set of Bounded Continuous functions from the real line onto itself. Think of it as a linear subspace of $L^\infty(\mathbb{R})$. Define $A\colon CB(\mathbb{R}) \to \mathbb{R}$ by $A(f) = f(0)$.  This is a continuous linear function and $\|A\|=1$. By the Hahn-Banach Theorem, $A$ can be extended to $A'\colon L^\infty(\mathbb{R})\to \mathbb{R} $ preserving the norm, i.e. $\|A'\|=1$. The question is: Can $A'$ be represented by some $g \in L^1(\mathbb{R})$ ? That is: Can we find  $g \in L^1(\mathbb{R})$ such that for all $f \in L^\infty(\mathbb{R})$ the following equality holds:
$$
A'(f) = \int_\mathbb{R} f\cdot g \, d\lambda
$$
All i know so far is that if that $g$ exists then it must satisfy $\|g\|_1\geq 1 $ beacuse $T(f) \leq \|g\|_1\cdot\|f\|_\infty$.
Any help would be appreciated.
 A: There is no $g \in L^1(\mathbb{R})$ such that for all $f \in L^\infty(\mathbb{R})$ the following equality holds:
$$
A'(f) = \int_\mathbb{R} f\cdot g \, d\lambda
$$
Proof:
Suppose such $g$ exists.
For $n \in \Bbb N$, let $f_n$ be a continuous bounded function such that:

*

*for all $x \in \Bbb R $, $0 \leqslant f_n(x)\leqslant 1$;

*$ \textrm{supp} f_n \subseteq [-\frac{1}{n+1}, \frac{1}{n+1}] $;

*$f_n(0)=1$.

Then, for all $n \in \Bbb N$, we have
$$1=|f_n(0)| = |A(f_n)| = |A'(f_n)|= \left | \int_{\Bbb R} f \cdot g \,d\lambda \right | \leqslant  \int_{\Bbb R}| f |\cdot | g |\,d\lambda  \leqslant  \int_{\Bbb R} \chi_{[-\frac{1}{n+1}, \frac{1}{n+1}]}\cdot | g |\,d\lambda $$
So, for all $n \in \Bbb N$, we have
$$1  \leqslant  \int_{\Bbb R} \chi_{[-\frac{1}{n+1}, \frac{1}{n+1}]}\cdot | g |\,d\lambda \tag{1}$$
Now, note that $\chi_{[-\frac{1}{n+1}, \frac{1}{n+1}]}\cdot | g |$ converge pointwise to $0$ a.e.. Since  $g \in L^1(\mathbb{R})$ and, for all $n \in \Bbb N$, $\chi_{[-\frac{1}{n+1}, \frac{1}{n+1}]}\cdot | g | \leqslant |g|$, we have, by Lebesgue Dominated Convergence Theorem, that:
$$ \lim_n \int_{\Bbb R} \chi_{[-\frac{1}{n+1}, \frac{1}{n+1}]}\cdot | g |\,d\lambda = \int_{\Bbb R} \lim_n  \chi_{[-\frac{1}{n+1}, \frac{1}{n+1}]}\cdot | g |\,d\lambda =  \int_{\Bbb R} 0 \, d\lambda =0$$
Contradiction to $(1)$.
