The dual space of L-infinity

Abstract

In this report we examine the dual space of $\ell^\infty$. If $p \in [1,\infty)$ and $q \in [1,\infty]$ satisfy $\frac{1}{p}+\frac{1}{q}=1$, then one can identify the spaces $\ell^q$ and $(\ell^p)'$ in a natural way via an isometric isomorphism. This identification does not extend to the case $p=\infty$ and $q=1$. We prove that the obvious candidate for an isometric isomorphism from $\ell^1$ into $(\ell^\infty)'$ fails to be surjective, and moreover, that an isometric isomorphism (even a homeomorphism) between these spaces does not exist at all.

We introduce a space that we can identify with $(\ell^\infty)'$ via an isometric isomorphism. This is the space of bounded finitely additive measures on $\mathbb{N}$, denoted by $\ba(\mathbb{N}, \mathcal{P}(\mathbb{N}))$. Having found this characterization of $(\ell^\infty)'$, we examine what kinds of finitely additive measures on $\mathbb{N}$ exist. These include $\sigma$-additive measures that are induced by $\ell^1$, diffuse measures, shift-invariant and more general invariant measures, measures that extend the asymptotic density, $0,1$-valued measures and stretchable, thinnable and elastic measures. Elastic measures can be considered the nicest measures on $\mathbb{N}$, from an intuitive point of view.

We also describe the functionals that correspond to particular types of measures and vice versa. Moreover, we prove that the collection of ultrafilters on $\mathbb{N}$ can be identified with the collection of $0,1$-valued measures on $\mathbb{N}$, which, in turn, can be identified with the collection of multiplicative functionals on $\ell^\infty$.