= Solution
On $C_c^0(\mathbb R^n)$ equipped with the $L^\infty$ norm, point evaluation is linear and satisfies
$$
|\Lambda f|=|f(0)|\leq\|f\|_\infty,
$$
with equality for a compactly supported function equal to one near zero. Thus $\|\Lambda\|=1$. The <Hahn-Banach theorem> extends it, without increasing its norm, to a continuous functional $\widetilde\Lambda$ on $L^\infty(\mathbb R^n)$.
This extension cannot be represented by an $L^1$ function. If $h\in L^1$ satisfied $\widetilde\Lambda(f)=\int fh$, choose continuous functions $0\leq f_k\leq1$ with $f_k(0)=1$ and support in the ball of radius $1/k$. Then
$$
1=\widetilde\Lambda(f_k)=\int f_kh\longrightarrow0
$$
by the <absolute continuity of the Lebesgue integral>, a contradiction. Therefore $(L^\infty)'$ strictly contains the functionals represented by $L^1$; this is a <singular functional on L infinity>.
Finally, $L^p(\mathbb R^n)$ is a <reflexive Banach space> exactly when
$$
1<p<\infty.
$$
The endpoint spaces $L^1$ and $L^\infty$ are not reflexive.
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