A Schauder basis for \(L_2\) consisting of non-negative functions

Abstract

We prove that \(L_2(\mathbb {R})\) contains a Schauder basis of non-negative functions. Similarly, for all \(1<p<\infty \), \(L_{p}(\mathbb {R})\) contains a Schauder basic sequence of non-negative functions such that \(L_p(\mathbb {R})\) embeds into the closed span of the sequence. We prove as well that if X is a separable Banach space with the bounded approximation property, then any set in X with dense span contains a quasi-basis (Schauder frame) for X.

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Acknowledgements

We thank the anonymous referee for their careful reading of the paper and helpful comments. The second author gratefully acknowledges the hospitality and support of the Academia Sinica Institute of Mathematics (Taipei, Taiwan).

Funding

Daniel Freeman was supported by Grants 353293 and 706481 from the Simons Foundation. Alexander M. Powell was supported by NSF DMS Grant 1521749.

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Freeman, D., Powell, A.M. & Taylor, M.A. A Schauder basis for \(L_2\) consisting of non-negative functions. Math. Ann. (2021). https://doi.org/10.1007/s00208-021-02143-4

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Mathematics Subject Classification

  • 46B03
  • 46B15
  • 46E30
  • 42C15