Advances in Operator Theory

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

Gilles Godefroy and Nicolas Lerner

Full-text: Open access


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.

Article information

Adv. Oper. Theory, Volume 3, Number 1 (2018), 61-74.

Received: 20 February 2017
Accepted: 14 April 2017
First available in Project Euclid: 5 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46B25: Classical Banach spaces in the general theory
Secondary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

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


Godefroy, Gilles; Lerner, Nicolas. Some natural subspaces and quotient spaces of $L^1$. Adv. Oper. Theory 3 (2018), no. 1, 61--74. doi:10.22034/aot.1702-1124.

Export citation


  • R. A. Adams and J. J. F. Fournier, Sobolev spaces, second ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003.
  • D. E. Alspach, Quotients of $c_{0}$ are almost isometric to subspaces of $c_{0}$, Proc. Amer. Math. Soc. 76 (1979), no. 2, 285–288.
  • J. Boclé, Sur la théorie ergodique, Ann. Inst. Fourier (Grenoble) 10 (1960), 1–45.
  • P. G. Casazza and N. J. Kalton, Notes on approximation properties in separable Banach spaces, Geometry of Banach spaces (Strobl, 1989), London Math. Soc. Lecture Note Ser., vol. 158, Cambridge Univ. Press, Cambridge, 1990, pp. 49–63.
  • M. Cúth, O. Kalenda, and P. Kaplický, Isometric representation of Lipschitz-free spaces over convex domains in finite-dimensional spaces, arXiv: 1610.03966.
  • M. Cúth, M. Doucha, and P. Wojtaszczyk, On the structure of Lipschitz-free spaces, Proc. Amer. Math. Soc. 144 (2016), no. 9, 3833–3846.
  • G. de Rham, Variétés différentiables. Formes, courants, formes harmoniques, Hermann, Paris, 1973, Troisième édition revue et augmentée, Publications de l'Institut de Mathématique de l'Université de Nancago, III, Actualités Scientifiques et Industrielles, No. 1222b.
  • L. Gaillard and P. Lefèvre, Lacunary Müntz spaces: isomorphisms and Carleson embeddings, arXiv: 1701.05807v1.
  • A. Godard, Tree metrics and their Lipschitz-free spaces, Proc. Amer. Math. Soc. 138 (2010), no. 12, 4311–4320.
  • G. Godefroy and N. J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (2003), no. 1, 121–141, Dedicated to Professor Aleksander Pelczynski on the occasion of his 70th birthday.
  • G. Godefroy, N. J. Kalton, and D. Li, On subspaces of $L^1$ which embed into $l_1$, J. Reine Angew. Math. 471 (1996), 43–75.
  • ––––, Operators between subspaces and quotients of $L^1$, Indiana Univ. Math. J. 49 (2000), no. 1, 245–286.
  • G. Godefroy, G. Lancien, and V. Zizler, The non-linear geometry of Banach spaces after Nigel Kalton, Rocky Mountain J. Math. 44 (2014), no. 5, 1529–1583.
  • G. Godefroy, Sous-espaces bien disposés de $L^{1}$-applications, Trans. Amer. Math. Soc. 286 (1984), no. 1, 227–249.
  • ––––, On Riesz subsets of abelian discrete groups, Israel J. Math. 61 (1988), no. 3, 301–331.
  • ––––, Unconditionality in spaces of smooth functions, Arch. Math. (Basel) 92 (2009), no. 5, 476–484.
  • G. Godefroy and D. Li, Some natural families of $M$-ideals, Math. Scand. 66 (1990), no. 2, 249–263.
  • G. Godefroy and M. Talagrand, Classes d'espaces de Banach à prédual unique, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 5, 323–325.
  • V. Gurariĭ and V. I. Macaev, Lacunary power sequences in spaces $C$ and $L_{p}$, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 3–14.
  • V. I. Gurariy and W. Lusky, Geometry of Müntz spaces and related questions, Lecture Notes in Mathematics, vol. 1870, Springer-Verlag, Berlin, 2005.
  • P. Harmand, D. Werner, and W. Werner, $M$-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993.
  • S. Hellerstein, Some analytic varieties in the polydisc and the Müntz-Szasz problem in several variables, Trans. Amer. Math. Soc. 158 (1971), 285–292.
  • J. Horváth, Topological vector spaces and distributions, Vol. I, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966.
  • R. C. James, Renorming the Banach space $c_{0}$, Proc. Amer. Math. Soc. 80 (1980), no. 4, 631–634.
  • M. A. Japón Pineda and C. Lennard, Second dual projection characterizations of three classes of $L_0$-closed, convex, bounded sets in $L_1$, J. Math. Anal. Appl. 342 (2008), no. 1, 1–16.
  • W. B. Johnson and M. Zippin, Subspaces and quotient spaces of $(\sum G_{n})_{l_{p}}$ and $(\sum G_{n})_{c_{0}}$, Israel J. Math. 17 (1974), 50–55.
  • N. J. Kalton, Extension of linear operators and Lipschitz maps into $C(K)$-spaces, New York J. Math. 13 (2007), 317–381.
  • N. J. Kalton and Dirk Werner, Property $(M)$, $M$-ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995), 137–178.
  • P. L. Kaufmann, Products of Lipschitz-free spaces and applications, Studia Math. 226 (2015), no. 3, 213–227.
  • C. M. Kootwijk and F. P. T. Baaijens, Application of the satisfying Babuška-Brezzi method to a two-dimensional diffusion problem, Appl. Math. Modelling 17 (1993), no. 4, 184–194.
  • G. Lancien and E. Pernecká, Approximation properties and Schauder decompositions in Lipschitz-free spaces, J. Funct. Anal. 264 (2013), no. 10, 2323–2334.
  • N. Lerner, A note on Lipschitz spaces, preprint.
  • J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin-New York, 1977, Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92.
  • H. Pfitzner, Separable L-embedded Banach spaces are unique preduals, Bull. Lond. Math. Soc. 39 (2007), no. 6, 1039–1044.
  • N. Weaver, On the unique predual problem for Lipschitz spaces, arXiv: 1611.01812v2.
  • ––––, Lipschitz algebras, World Scientific Publishing Co., Inc., River Edge, NJ, 1999.