Godefroy, G., Lerner, N. (2018). Some natural subspaces and quotient spaces of $L^1$. Advances in Operator Theory, 3(1), 61-74. doi: 10.22034/aot.1702-1124

Gilles Godefroy; Nicolas Lerner. "Some natural subspaces and quotient spaces of $L^1$". Advances in Operator Theory, 3, 1, 2018, 61-74. doi: 10.22034/aot.1702-1124

Godefroy, G., Lerner, N. (2018). 'Some natural subspaces and quotient spaces of $L^1$', Advances in Operator Theory, 3(1), pp. 61-74. doi: 10.22034/aot.1702-1124

Godefroy, G., Lerner, N. Some natural subspaces and quotient spaces of $L^1$. Advances in Operator Theory, 2018; 3(1): 61-74. doi: 10.22034/aot.1702-1124

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

Receive Date: 20 February 2017,
Accept Date: 20 February 2017

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 identically. 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 dilation-stable 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.