Abstract
In this chapter, we study the sum L 1 +L ∞ of the spaces L 1 and L ∞ . We show that L 1 +L ∞ equipped with a natural norm \(\|\cdot \|_{\mathbf{L}_{1}+\mathbf{L}_{\infty }}\) is a symmetric space. The norm \(\|\cdot \|_{\mathbf{L}_{1}+\mathbf{L}_{\infty }}\) can be written in the form ∫ 0 1 f ∗ dm using the maximal property of decreasing rearrangements f ∗. We also describe embeddings of L 1 and L ∞ into L 1 +L ∞ and the closure R 0 of L 1 in L 1 +L ∞ .
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Rubshtein, BZ.A., Grabarnik, G.Y., Muratov, M.A., Pashkova, Y.S. (2016). The Space L 1 +L ∞ . In: Foundations of Symmetric Spaces of Measurable Functions. Developments in Mathematics, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-319-42758-4_4
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DOI: https://doi.org/10.1007/978-3-319-42758-4_4
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