Abstract

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.