𝐿_{𝑝}+𝐿_{∞} and 𝐿_{𝑝}∩𝐿_{∞} are not isomorphic for all 1≤𝑝<∞, 𝑝≠2

Abstract

We prove the result stated in the title. It comes as a consequence of the fact that the space L p ∩ L ∞ L_p \cap L_{\infty } , 1 ≤ p > ∞ 1\leq p>\infty , p ≠ 2 p\neq 2 , does not contain a complemented subspace isomorphic to L p L_p . In particular, as a subproduct, we show that L p ∩ L ∞ L_p \cap L_{\infty } contains a complemented subspace isomorphic to ℓ 2 {\ell }_2 if and only if p = 2 p = 2 .