Duality and distance formulas in Banach function spaces

Abstract

We consider pairs of non reflexive Banach spaces $(E_0, E)$ such that $E_0$ is defined in terms of a little-o condition and E is defined by the corresponding big-O condition. Under suitable assumptions on the pair $(E_0, E)$ there exists a reflexive and separable Banach space X (in which E is continuously embedded and dense) naturally associated to E which characterizes quantitatively weak compactness of bounded linear operators

$$\begin{aligned} T: E_0 \rightarrow Z \end{aligned}$$

where Z is an arbitrary Banach space. Pairs include (VMO, BMO), where BMO is the space of John-Nirenberg, $(B_0, B)$ where B is a recently introduced space by Bourgain-Brezis-Mironescu ([6]) and some Orlicz pairs $(L^{\psi }_0, L^{\psi })$ where $L^{\psi }_0$ is the closure of $L^\infty$ in the Orlicz space $L^{\psi }$, Marcinkiewicz pairs $(L^{q, \infty }_0, L^{q, \infty })$ where $L^{q, \infty }_0$ is the closure of $L^\infty$ in the Marcinkiewicz weak–$L^q$ denoted by $L^{q, \infty }$. More generally, Banach function spaces are considered. The main results are duality formulas of the type

$$\begin{aligned} E_0^{**}&\simeq E\qquad \text {isometrically}\end{aligned}$$

(1)

$$\begin{aligned}E^{*}&\simeq E_0^*\oplus _1 E_0^\perp\end{aligned}$$

(2)

and distance formulas.