riesz representation theorem for the case when p is infinity for the case of lebesgue measure

Question

can we extend Riesz representation theorem for the case when p is infinity? If not can you give me a counter example?

Answer 1

No, it can not be extended to $p=\infty$. Counterexamples can be constructed as follows. Let $\ell^\infty$ be the space of bounded real sequences $(x_1,x_2,x_3,\ldots)$, and let $V$ be the subspace of convergent sequences. Define $L:V \to \mathbb{R}$ by $L((x_1,x_2,x_3,\ldots)) = \lim\limits_{n\to\infty} x_n$. Then $L$ is a linear functional on $V$ with norm $\| L \|=1$. By the Hahn-Banach theorem there exists an extension of $L$ to a linear functional $\bar{L} : \ell^\infty \to \mathbb{R}$ with norm $\|\bar{L}\|=1$. It is straightforward to show that $L$ is not represented by an element of $\ell^1$.