Some natural subspaces and quotient spaces of $L^1$

Abstract

We show that the space ${\text{Lip}}_0({\mathbb{R}}^n)$ is the dual space of $L^1({\mathbb{R}}^n;{\mathbb{R}}^n)/N$ where $N$ is the subspace of $L^1({\mathbb{R}}^n;{\mathbb{R}}^n)$ consisting of vector fields whose divergence vanishes. We prove that although the quotient space $L^1({\mathbb{R}}^n;{\mathbb{R}}^n)/N$ is weakly sequentially complete, the subspace $N$ is not nicely placed - in other words, its unit ball is not closed for the topology ${\tau }_m$ of local convergence in measure. We prove that if $\Omega$ is a bounded open star-shaped subset of ${\mathbb{R}}^n$ and $X$ is a closed subspace of $L^1(\Omega )$ consisting of continuous functions, then the unit ball of $X$ is compact for the compact-open topology on $\Omega$. It follows in particular that such spaces $X$, when they have Grothendieck's approximation property, have unconditional finite-dimensional decompositions and are isomorphic to weak*-closed subspaces of $l^1$. Numerous examples are provided where such results apply.

13 pages

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