Abstract
In this chapter, we prove the main embedding theorem for symmetric spaces. The theorem asserts that for every symmetric space X, there are continuous embeddings L 1 ∩L ∞ ⊆ X ⊆ L 1 +L ∞ and inequalities \(2\| \cdot \|_{\mathbf{L}_{1}\cap \mathbf{L}_{\infty }} \geq (\varphi _{\mathbf{X}})^{-1}(1) \times \|\cdot \|_{\mathbf{X}} \geq\) \(\geq \|\cdot \|_{\mathbf{L}_{1}+\mathbf{L}_{\infty }}.\) The space L 0 of all measurable functions and the embedding L 1 +L ∞ ⊂ L 0 are also considered.
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© 2016 Springer International Publishing Switzerland
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Rubshtein, BZ.A., Grabarnik, G.Y., Muratov, M.A., Pashkova, Y.S. (2016). Embeddings L1 ∩ L∞ ⊆ X ⊆ L1 + L∞ ⊆ L0 . 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_5
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DOI: https://doi.org/10.1007/978-3-319-42758-4_5
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-42756-0
Online ISBN: 978-3-319-42758-4
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