We use the compactness theorem of continuous logic to give a new proof that \(L^r([0,1]; {\mathbb {R}})\) isometrically embeds into \(L^p([0,1]; {\mathbb {R}})\) whenever \(1 \le p \le r \le 2\). We will also give a proof for the complex case. This will involve a new characterization of complex \(L^p\) spaces based on Banach lattices.

中文翻译：

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Lebesgue空间的连续逻辑和嵌入

我们使用连续逻辑的紧致性定理给出一个新证明，证明\（L ^ r（[0,1]; {\ mathbb {R}}）\）等距地嵌入\（L ^ p（[0,1] ; {\ mathbb {R}}）\）每当\（1 \ le p \ le r \ le 2 \）。我们还将为复杂的情况提供证明。这将涉及基于Banach格的复杂\（L ^ p \）空间的新表征。