Mathlib.MeasureTheory.Function.ContinuousMapDense Mathlib.Analysis.Complex.Periodic Mathlib.Analysis.Normed.Algebra.Norm Mathlib.Computability.AkraBazzi.GrowsPolynomially Mathlib.Geometry.Manifold.ContMDiffMap Mathlib.Algebra.MvPolynomial.Funext Mathlib.NumberTheory.NumberField.Norm Mathlib.Analysis.Convex....
Proof: Since the norm operator \parallel\cdot\parallel_u on L^u(X) for any u \in [1, \infty) is absolutely homogenous, we can assume WLOG that \parallel f\parallel_p = \parallel g\parallel_q = 1 . With monotonicity of the integration operator, we have by Young's inequality, ...