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Winter 2018 Some natural subspaces and quotient spaces of $L^1$
Gilles Godefroy, Nicolas Lerner
Adv. Oper. Theory 3(1): 61-74 (Winter 2018). DOI: 10.22034/aot.1702-1124


We show that the space $\mathrm{Lip}_0(\mathbb R^n)$ 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.


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Gilles Godefroy. Nicolas Lerner. "Some natural subspaces and quotient spaces of $L^1$." Adv. Oper. Theory 3 (1) 61 - 74, Winter 2018.


Received: 20 February 2017; Accepted: 14 April 2017; Published: Winter 2018
First available in Project Euclid: 5 December 2017

zbMATH: 06804317
MathSciNet: MR3730340
Digital Object Identifier: 10.22034/aot.1702-1124

Primary: 46B25
Secondary: 46E30

Keywords: Lipschitz-free spaces over $\mathbb{R}^n$ , nicely placed subspaces of $L^1$ , subspaces of $l^1$

Rights: Copyright © 2018 Tusi Mathematical Research Group


Vol.3 • No. 1 • Winter 2018
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