Geodesic
On Quasibases and Bases of Symmetric Spaces Consisting of Nonnegative Functions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 20-28

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Based on the study of the geometric properties of unconditional quasibasic sequences, we show that in an arbitrary symmetric space there exists no unconditional quasibasis consisting of nonnegative functions. Moreover, we demonstrate that in an arbitrary Banach function lattice $X$ of type $p>1$ one can introduce an equivalent norm such that there exists no monotone (with respect to the new norm) basis in $X$ that consists of nonnegative functions.
Keywords: basis, quasibasis, basic sequence, symmetric space, Rademacher system, type of a Banach space. 
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     author = {S. V. Astashkin and P. A. Terekhin},
     title = {On {Quasibases} and {Bases} of {Symmetric} {Spaces} {Consisting} of {Nonnegative} {Functions}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     year = {2022},
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S. V. Astashkin; P. A. Terekhin. On Quasibases and Bases of Symmetric Spaces Consisting of Nonnegative Functions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 20-28. http://geodesic.mathdoc.fr/item/TM_2022_319_a1/